| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 36·6-s + 49·7-s + 64·8-s − 162·9-s − 187·11-s + 144·12-s − 627·13-s + 196·14-s + 256·16-s − 1.81e3·17-s − 648·18-s + 258·19-s + 441·21-s − 748·22-s − 2.97e3·23-s + 576·24-s − 2.50e3·26-s − 3.64e3·27-s + 784·28-s + 1.29e3·29-s + 1.91e3·31-s + 1.02e3·32-s − 1.68e3·33-s − 7.25e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.465·11-s + 0.288·12-s − 1.02·13-s + 0.267·14-s + 1/4·16-s − 1.52·17-s − 0.471·18-s + 0.163·19-s + 0.218·21-s − 0.329·22-s − 1.17·23-s + 0.204·24-s − 0.727·26-s − 0.962·27-s + 0.188·28-s + 0.286·29-s + 0.358·31-s + 0.176·32-s − 0.269·33-s − 1.07·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - p^{2} T + p^{5} T^{2} \) |
| 11 | \( 1 + 17 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 627 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1813 T + p^{5} T^{2} \) |
| 19 | \( 1 - 258 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2970 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1299 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1916 T + p^{5} T^{2} \) |
| 37 | \( 1 + 6578 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6676 T + p^{5} T^{2} \) |
| 43 | \( 1 + 3178 T + p^{5} T^{2} \) |
| 47 | \( 1 - 22001 T + p^{5} T^{2} \) |
| 53 | \( 1 + 26168 T + p^{5} T^{2} \) |
| 59 | \( 1 - 3932 T + p^{5} T^{2} \) |
| 61 | \( 1 + 48740 T + p^{5} T^{2} \) |
| 67 | \( 1 - 44832 T + p^{5} T^{2} \) |
| 71 | \( 1 - 63736 T + p^{5} T^{2} \) |
| 73 | \( 1 + 60470 T + p^{5} T^{2} \) |
| 79 | \( 1 + 43721 T + p^{5} T^{2} \) |
| 83 | \( 1 + 1172 p T + p^{5} T^{2} \) |
| 89 | \( 1 - 45560 T + p^{5} T^{2} \) |
| 97 | \( 1 - 57295 T + p^{5} T^{2} \) |
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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
−10.32087095076289790983231628339, −9.184108328565528352039135538973, −8.226368518473944405162280106681, −7.37510085643500339398729496814, −6.20208332352772434250961020667, −5.11562471425620453699527959507, −4.16046623414305192832407991512, −2.81592819782801204371212381755, −2.04452629479928993111322860698, 0,
2.04452629479928993111322860698, 2.81592819782801204371212381755, 4.16046623414305192832407991512, 5.11562471425620453699527959507, 6.20208332352772434250961020667, 7.37510085643500339398729496814, 8.226368518473944405162280106681, 9.184108328565528352039135538973, 10.32087095076289790983231628339
