| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 11-s + 12-s + 16-s + 17-s − 19-s + 22-s − 24-s − 27-s − 32-s − 33-s − 34-s + 38-s − 41-s − 2·43-s − 44-s + 48-s + 49-s + 51-s + 54-s − 57-s + 2·59-s + 64-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 11-s + 12-s + 16-s + 17-s − 19-s + 22-s − 24-s − 27-s − 32-s − 33-s − 34-s + 38-s − 41-s − 2·43-s − 44-s + 48-s + 49-s + 51-s + 54-s − 57-s + 2·59-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5885706700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5885706700\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.65590402682138368580540241866, −11.54471611586141704153561438984, −10.43588891424852406051840638960, −9.692338832122206333700075116861, −8.496836338518290837190879315347, −8.085541165554903831343454602727, −6.92226717026077750498288682652, −5.49656216305026105712573419563, −3.40106322552307567973286887374, −2.22404656721183176474639929806,
2.22404656721183176474639929806, 3.40106322552307567973286887374, 5.49656216305026105712573419563, 6.92226717026077750498288682652, 8.085541165554903831343454602727, 8.496836338518290837190879315347, 9.692338832122206333700075116861, 10.43588891424852406051840638960, 11.54471611586141704153561438984, 12.65590402682138368580540241866
