| L(s) = 1 | − 2·3-s − 2·5-s − 7-s + 9-s + 2·11-s + 4·13-s + 4·15-s + 6·17-s + 2·21-s − 25-s + 4·27-s + 2·29-s − 4·31-s − 4·33-s + 2·35-s − 8·39-s + 6·41-s − 6·43-s − 2·45-s + 49-s − 12·51-s − 12·53-s − 4·55-s − 10·59-s + 2·61-s − 63-s − 8·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 1.03·15-s + 1.45·17-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s + 0.338·35-s − 1.28·39-s + 0.937·41-s − 0.914·43-s − 0.298·45-s + 1/7·49-s − 1.68·51-s − 1.64·53-s − 0.539·55-s − 1.30·59-s + 0.256·61-s − 0.125·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.91514929648448, −12.51830898871050, −12.10528947107054, −11.74686897665321, −11.35503318424093, −10.89790754852444, −10.54842825497156, −9.975336337214937, −9.414092098853447, −8.980260795127164, −8.365133541977800, −7.842738374361592, −7.553840182418126, −6.813046092111488, −6.338129081431323, −6.055139626484727, −5.504300154044657, −4.998437472739846, −4.392353098083263, −3.853238850212310, −3.344139739187297, −3.017075507387276, −1.897641630118187, −1.230040224048262, −0.7093854012051795, 0,
0.7093854012051795, 1.230040224048262, 1.897641630118187, 3.017075507387276, 3.344139739187297, 3.853238850212310, 4.392353098083263, 4.998437472739846, 5.504300154044657, 6.055139626484727, 6.338129081431323, 6.813046092111488, 7.553840182418126, 7.842738374361592, 8.365133541977800, 8.980260795127164, 9.414092098853447, 9.975336337214937, 10.54842825497156, 10.89790754852444, 11.35503318424093, 11.74686897665321, 12.10528947107054, 12.51830898871050, 12.91514929648448
