| L(s) = 1 | − 3-s − 4-s + 3·7-s + 12-s − 2·13-s − 2·19-s − 3·21-s − 25-s − 3·28-s − 2·31-s − 2·37-s + 2·39-s − 2·43-s + 6·49-s + 2·52-s + 2·57-s + 4·61-s − 2·67-s − 2·73-s + 75-s + 2·76-s + 3·79-s + 3·84-s − 6·91-s + 2·93-s + 3·97-s + 100-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 3·7-s + 12-s − 2·13-s − 2·19-s − 3·21-s − 25-s − 3·28-s − 2·31-s − 2·37-s + 2·39-s − 2·43-s + 6·49-s + 2·52-s + 2·57-s + 4·61-s − 2·67-s − 2·73-s + 75-s + 2·76-s + 3·79-s + 3·84-s − 6·91-s + 2·93-s + 3·97-s + 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1536087529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1536087529\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 61 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 37 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.485956469567913178050931832035, −9.183085154774233649556717769352, −8.842317718927716508914612436927, −8.653415428450603363243827394560, −8.460463491422585262538480959504, −8.381466695913539299428422064099, −7.77816661977831245328847855417, −7.71289172270997632549788740847, −7.40839936932535004014219930441, −7.07862203409342405010542640383, −6.93144257656704403080897475928, −6.27253806498845619199978051490, −6.25840531422963715206156536175, −5.46203302444493423582000358701, −5.31702215179182313684449610446, −5.27145548903707579677365037467, −5.11174742429741760550324781207, −4.60438739393588900103921903461, −4.26186758995398828167800933452, −4.21342677716110972285058748637, −3.73326539621520962191708937154, −3.11406532930362926499783686517, −2.11251334559918474214562472631, −2.10487563753653106381549000775, −1.77380266066447465432251107827,
1.77380266066447465432251107827, 2.10487563753653106381549000775, 2.11251334559918474214562472631, 3.11406532930362926499783686517, 3.73326539621520962191708937154, 4.21342677716110972285058748637, 4.26186758995398828167800933452, 4.60438739393588900103921903461, 5.11174742429741760550324781207, 5.27145548903707579677365037467, 5.31702215179182313684449610446, 5.46203302444493423582000358701, 6.25840531422963715206156536175, 6.27253806498845619199978051490, 6.93144257656704403080897475928, 7.07862203409342405010542640383, 7.40839936932535004014219930441, 7.71289172270997632549788740847, 7.77816661977831245328847855417, 8.381466695913539299428422064099, 8.460463491422585262538480959504, 8.653415428450603363243827394560, 8.842317718927716508914612436927, 9.183085154774233649556717769352, 9.485956469567913178050931832035
