| L(s) = 1 | + 2-s + 4-s − 2·5-s − 3·7-s + 8-s − 2·10-s − 5·11-s − 3·13-s − 3·14-s + 16-s − 3·17-s + 5·19-s − 2·20-s − 5·22-s − 3·23-s − 25-s − 3·26-s − 3·28-s + 4·31-s + 32-s − 3·34-s + 6·35-s + 37-s + 5·38-s − 2·40-s − 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.13·7-s + 0.353·8-s − 0.632·10-s − 1.50·11-s − 0.832·13-s − 0.801·14-s + 1/4·16-s − 0.727·17-s + 1.14·19-s − 0.447·20-s − 1.06·22-s − 0.625·23-s − 1/5·25-s − 0.588·26-s − 0.566·28-s + 0.718·31-s + 0.176·32-s − 0.514·34-s + 1.01·35-s + 0.164·37-s + 0.811·38-s − 0.316·40-s − 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.19101722664861990638377585528, −9.388240731285929266205934536593, −8.003115280121563730110945330920, −7.46068180666953821854598915657, −6.47610998714696152245554562179, −5.42150318427562977205146477273, −4.48939808970851696949242760762, −3.36955088619499482554151547955, −2.53768371991226370428359299532, 0,
2.53768371991226370428359299532, 3.36955088619499482554151547955, 4.48939808970851696949242760762, 5.42150318427562977205146477273, 6.47610998714696152245554562179, 7.46068180666953821854598915657, 8.003115280121563730110945330920, 9.388240731285929266205934536593, 10.19101722664861990638377585528
