| L(s) = 1 | + 2-s + 4-s − 4·5-s − 7-s + 8-s − 4·10-s + 11-s − 3·13-s − 14-s + 16-s − 3·17-s − 5·19-s − 4·20-s + 22-s − 5·23-s + 11·25-s − 3·26-s − 28-s − 4·29-s − 10·31-s + 32-s − 3·34-s + 4·35-s − 37-s − 5·38-s − 4·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s + 0.353·8-s − 1.26·10-s + 0.301·11-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 1.14·19-s − 0.894·20-s + 0.213·22-s − 1.04·23-s + 11/5·25-s − 0.588·26-s − 0.188·28-s − 0.742·29-s − 1.79·31-s + 0.176·32-s − 0.514·34-s + 0.676·35-s − 0.164·37-s − 0.811·38-s − 0.632·40-s + 0.937·41-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 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.34656449179909477685318487264, −9.100924645180121051837481922239, −8.162699398202549566838232688366, −7.32327856135543720727953115667, −6.66446396406084609826772404942, −5.36951137817095598841912521526, −4.13511654126083004672518336782, −3.82521452982993692459318044646, −2.37866343224007325240428184274, 0,
2.37866343224007325240428184274, 3.82521452982993692459318044646, 4.13511654126083004672518336782, 5.36951137817095598841912521526, 6.66446396406084609826772404942, 7.32327856135543720727953115667, 8.162699398202549566838232688366, 9.100924645180121051837481922239, 10.34656449179909477685318487264
