| L(s) = 1 | + 3.53·2-s + 4.46·4-s + 5·5-s − 12.4·8-s + 17.6·10-s + 2.93·11-s + 19.0·13-s − 79.7·16-s + 122.·17-s − 107.·19-s + 22.3·20-s + 10.3·22-s − 210.·23-s + 25·25-s + 67.3·26-s − 95.4·29-s + 94.3·31-s − 181.·32-s + 432.·34-s + 97.1·37-s − 379.·38-s − 62.3·40-s − 491.·41-s − 43.0·43-s + 13.1·44-s − 743.·46-s + 473.·47-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.558·4-s + 0.447·5-s − 0.551·8-s + 0.558·10-s + 0.0805·11-s + 0.406·13-s − 1.24·16-s + 1.74·17-s − 1.29·19-s + 0.249·20-s + 0.100·22-s − 1.90·23-s + 0.200·25-s + 0.507·26-s − 0.611·29-s + 0.546·31-s − 1.00·32-s + 2.18·34-s + 0.431·37-s − 1.61·38-s − 0.246·40-s − 1.87·41-s − 0.152·43-s + 0.0449·44-s − 2.38·46-s + 1.46·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3.53T + 8T^{2} \) |
| 11 | \( 1 - 2.93T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 95.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 97.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 491.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 43.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 473.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 760.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 309.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 665.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 24.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 406.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−8.303051664168433483619884952271, −7.41075753999081805837649244010, −6.23158719810207553787429696299, −5.97264994613631742559995850922, −5.13052200273767844145512290324, −4.19128463776645862515108061213, −3.57217315147426543463037173250, −2.58473619934838414296704277849, −1.54223710888771067470345598373, 0,
1.54223710888771067470345598373, 2.58473619934838414296704277849, 3.57217315147426543463037173250, 4.19128463776645862515108061213, 5.13052200273767844145512290324, 5.97264994613631742559995850922, 6.23158719810207553787429696299, 7.41075753999081805837649244010, 8.303051664168433483619884952271
