| L(s) = 1 | + 4·2-s − 17·3-s + 16·4-s + 25·5-s − 68·6-s − 49·7-s + 64·8-s + 46·9-s + 100·10-s − 715·11-s − 272·12-s + 331·13-s − 196·14-s − 425·15-s + 256·16-s − 1.69e3·17-s + 184·18-s − 1.71e3·19-s + 400·20-s + 833·21-s − 2.86e3·22-s − 3.95e3·23-s − 1.08e3·24-s + 625·25-s + 1.32e3·26-s + 3.34e3·27-s − 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.09·3-s + 1/2·4-s + 0.447·5-s − 0.771·6-s − 0.377·7-s + 0.353·8-s + 0.189·9-s + 0.316·10-s − 1.78·11-s − 0.545·12-s + 0.543·13-s − 0.267·14-s − 0.487·15-s + 1/4·16-s − 1.42·17-s + 0.133·18-s − 1.09·19-s + 0.223·20-s + 0.412·21-s − 1.25·22-s − 1.55·23-s − 0.385·24-s + 1/5·25-s + 0.384·26-s + 0.884·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 17 T + p^{5} T^{2} \) |
| 11 | \( 1 + 65 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 331 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1699 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1718 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3950 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4579 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6756 T + p^{5} T^{2} \) |
| 37 | \( 1 + 16518 T + p^{5} T^{2} \) |
| 41 | \( 1 - 18876 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2258 T + p^{5} T^{2} \) |
| 47 | \( 1 + 537 T + p^{5} T^{2} \) |
| 53 | \( 1 + 10984 T + p^{5} T^{2} \) |
| 59 | \( 1 + 25956 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39188 T + p^{5} T^{2} \) |
| 67 | \( 1 - 4416 T + p^{5} T^{2} \) |
| 71 | \( 1 + 31880 T + p^{5} T^{2} \) |
| 73 | \( 1 + 5018 T + p^{5} T^{2} \) |
| 79 | \( 1 + 27977 T + p^{5} T^{2} \) |
| 83 | \( 1 - 37644 T + p^{5} T^{2} \) |
| 89 | \( 1 + 17216 T + p^{5} T^{2} \) |
| 97 | \( 1 + 63175 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.09652400272546139287553037240, −12.20474211048146039768669001451, −10.90489649720915525320862384296, −10.31454653103567128053300563467, −8.356868565812265830201066133941, −6.58176882243876995004289383651, −5.76896697862978829156813695113, −4.56518325556061705113140981503, −2.47623506593761376059885085138, 0,
2.47623506593761376059885085138, 4.56518325556061705113140981503, 5.76896697862978829156813695113, 6.58176882243876995004289383651, 8.356868565812265830201066133941, 10.31454653103567128053300563467, 10.90489649720915525320862384296, 12.20474211048146039768669001451, 13.09652400272546139287553037240
