| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 13-s − 14-s − 15-s − 16-s + 18-s + 19-s − 21-s − 22-s − 24-s + 26-s + 27-s + 29-s − 30-s − 33-s + 35-s − 37-s + 38-s + 39-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 13-s − 14-s − 15-s − 16-s + 18-s + 19-s − 21-s − 22-s − 24-s + 26-s + 27-s + 29-s − 30-s − 33-s + 35-s − 37-s + 38-s + 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.104479589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104479589\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.66006050704642387693778431685, −11.90640003627446290009196776778, −10.50986026962350427505055367915, −9.395961570678151597911095642621, −8.462703563808204563312201481605, −7.50191162411095175056825563793, −6.23929186676262284482826313392, −4.80148076508186204024724787181, −3.62146687729465515194234591820, −3.03996178045392287303024917369,
3.03996178045392287303024917369, 3.62146687729465515194234591820, 4.80148076508186204024724787181, 6.23929186676262284482826313392, 7.50191162411095175056825563793, 8.462703563808204563312201481605, 9.395961570678151597911095642621, 10.50986026962350427505055367915, 11.90640003627446290009196776778, 12.66006050704642387693778431685
