| L(s) = 1 | − 3·2-s + 4-s + 14·5-s − 7·7-s + 21·8-s − 42·10-s + 11·11-s + 2·13-s + 21·14-s − 71·16-s + 74·17-s + 14·20-s − 33·22-s + 148·23-s + 71·25-s − 6·26-s − 7·28-s − 26·29-s + 112·31-s + 45·32-s − 222·34-s − 98·35-s − 98·37-s + 294·40-s + 10·41-s + 208·43-s + 11·44-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s + 1.25·5-s − 0.377·7-s + 0.928·8-s − 1.32·10-s + 0.301·11-s + 0.0426·13-s + 0.400·14-s − 1.10·16-s + 1.05·17-s + 0.156·20-s − 0.319·22-s + 1.34·23-s + 0.567·25-s − 0.0452·26-s − 0.0472·28-s − 0.166·29-s + 0.648·31-s + 0.248·32-s − 1.11·34-s − 0.473·35-s − 0.435·37-s + 1.16·40-s + 0.0380·41-s + 0.737·43-s + 0.0376·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.381533640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381533640\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 - 148 T + p^{3} T^{2} \) |
| 29 | \( 1 + 26 T + p^{3} T^{2} \) |
| 31 | \( 1 - 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 98 T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 T + p^{3} T^{2} \) |
| 43 | \( 1 - 208 T + p^{3} T^{2} \) |
| 47 | \( 1 + 460 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 204 T + p^{3} T^{2} \) |
| 61 | \( 1 - 178 T + p^{3} T^{2} \) |
| 67 | \( 1 + 924 T + p^{3} T^{2} \) |
| 71 | \( 1 - 748 T + p^{3} T^{2} \) |
| 73 | \( 1 + 230 T + p^{3} T^{2} \) |
| 79 | \( 1 + 456 T + p^{3} T^{2} \) |
| 83 | \( 1 - 228 T + p^{3} T^{2} \) |
| 89 | \( 1 - 198 T + p^{3} T^{2} \) |
| 97 | \( 1 - 562 T + p^{3} T^{2} \) |
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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
−9.886398178248379544860494558359, −9.329199835206800537243620382934, −8.605538940027315713883353534933, −7.57620374228745966013744538783, −6.65841078082897294538706593515, −5.69459977783263275639729743975, −4.70921456100148149312023808814, −3.20298106283103105230625728271, −1.83983440378421350374546410015, −0.841711287666450173254551209289,
0.841711287666450173254551209289, 1.83983440378421350374546410015, 3.20298106283103105230625728271, 4.70921456100148149312023808814, 5.69459977783263275639729743975, 6.65841078082897294538706593515, 7.57620374228745966013744538783, 8.605538940027315713883353534933, 9.329199835206800537243620382934, 9.886398178248379544860494558359
