| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 9-s + 4·11-s + 2·12-s − 13-s − 4·16-s + 4·17-s + 2·18-s + 7·19-s + 8·22-s + 6·23-s − 5·25-s − 2·26-s + 27-s − 8·29-s + 31-s − 8·32-s + 4·33-s + 8·34-s + 2·36-s + 7·37-s + 14·38-s − 39-s − 41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s + 1.20·11-s + 0.577·12-s − 0.277·13-s − 16-s + 0.970·17-s + 0.471·18-s + 1.60·19-s + 1.70·22-s + 1.25·23-s − 25-s − 0.392·26-s + 0.192·27-s − 1.48·29-s + 0.179·31-s − 1.41·32-s + 0.696·33-s + 1.37·34-s + 1/3·36-s + 1.15·37-s + 2.27·38-s − 0.160·39-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.040937367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.040937367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.85957688614011467088708385407, −7.21211977446279360622600598854, −6.59580632937904738409413318392, −5.65759312661357432881879911220, −5.23907612885705334959577467821, −4.31592812203653306929802788010, −3.58429756201014766378777015393, −3.20956772096571120049798575246, −2.17540526569732010497996031213, −1.08351983542254918255207166891,
1.08351983542254918255207166891, 2.17540526569732010497996031213, 3.20956772096571120049798575246, 3.58429756201014766378777015393, 4.31592812203653306929802788010, 5.23907612885705334959577467821, 5.65759312661357432881879911220, 6.59580632937904738409413318392, 7.21211977446279360622600598854, 7.85957688614011467088708385407
