| L(s) = 1 | − 4·2-s − 11·3-s + 16·4-s + 25·5-s + 44·6-s − 64·8-s − 122·9-s − 100·10-s + 83·11-s − 176·12-s + 83·13-s − 275·15-s + 256·16-s + 177·17-s + 488·18-s + 2.08e3·19-s + 400·20-s − 332·22-s − 3.17e3·23-s + 704·24-s + 625·25-s − 332·26-s + 4.01e3·27-s − 8.68e3·29-s + 1.10e3·30-s − 1.63e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.705·3-s + 1/2·4-s + 0.447·5-s + 0.498·6-s − 0.353·8-s − 0.502·9-s − 0.316·10-s + 0.206·11-s − 0.352·12-s + 0.136·13-s − 0.315·15-s + 1/4·16-s + 0.148·17-s + 0.355·18-s + 1.32·19-s + 0.223·20-s − 0.146·22-s − 1.24·23-s + 0.249·24-s + 1/5·25-s − 0.0963·26-s + 1.05·27-s − 1.91·29-s + 0.223·30-s − 0.305·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 11 T + p^{5} T^{2} \) |
| 11 | \( 1 - 83 T + p^{5} T^{2} \) |
| 13 | \( 1 - 83 T + p^{5} T^{2} \) |
| 17 | \( 1 - 177 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2082 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3170 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8681 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1636 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4298 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2356 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8738 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3641 T + p^{5} T^{2} \) |
| 53 | \( 1 - 33268 T + p^{5} T^{2} \) |
| 59 | \( 1 - 30968 T + p^{5} T^{2} \) |
| 61 | \( 1 + 4560 T + p^{5} T^{2} \) |
| 67 | \( 1 - 564 p T + p^{5} T^{2} \) |
| 71 | \( 1 + 59304 T + p^{5} T^{2} \) |
| 73 | \( 1 - 8910 T + p^{5} T^{2} \) |
| 79 | \( 1 - 27589 T + p^{5} T^{2} \) |
| 83 | \( 1 + 67676 T + p^{5} T^{2} \) |
| 89 | \( 1 + 10700 T + p^{5} T^{2} \) |
| 97 | \( 1 + 65075 T + p^{5} T^{2} \) |
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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
−9.718630732747844487443198797241, −8.980408896753061286752499122384, −7.917172008401726517592761551187, −7.00212003881103708446590166396, −5.88333025273338520444841795442, −5.41600106176275251608110304790, −3.78469759178805718868597049906, −2.42937699591590937688979161759, −1.17030980464259065978557420782, 0,
1.17030980464259065978557420782, 2.42937699591590937688979161759, 3.78469759178805718868597049906, 5.41600106176275251608110304790, 5.88333025273338520444841795442, 7.00212003881103708446590166396, 7.917172008401726517592761551187, 8.980408896753061286752499122384, 9.718630732747844487443198797241
