| L(s) = 1 | − 1.67e7·2-s + 5.46e11·3-s + 2.81e14·4-s + 1.83e17·5-s − 9.17e18·6-s − 6.94e20·7-s − 4.72e21·8-s + 5.98e22·9-s − 3.07e24·10-s − 3.82e25·11-s + 1.53e26·12-s − 2.81e27·13-s + 1.16e28·14-s + 1.00e29·15-s + 7.92e28·16-s − 8.61e28·17-s − 1.00e30·18-s + 2.22e31·19-s + 5.16e31·20-s − 3.79e32·21-s + 6.42e32·22-s − 3.43e33·23-s − 2.58e33·24-s + 1.58e34·25-s + 4.72e34·26-s − 9.81e34·27-s − 1.95e35·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.11·3-s + 0.5·4-s + 1.37·5-s − 0.790·6-s − 1.36·7-s − 0.353·8-s + 0.250·9-s − 0.973·10-s − 1.17·11-s + 0.559·12-s − 1.43·13-s + 0.968·14-s + 1.53·15-s + 0.250·16-s − 0.0615·17-s − 0.176·18-s + 1.04·19-s + 0.688·20-s − 1.53·21-s + 0.828·22-s − 1.49·23-s − 0.395·24-s + 0.894·25-s + 1.01·26-s − 0.838·27-s − 0.684·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(25)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{51}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.67e7T \) |
| good | 3 | \( 1 - 5.46e11T + 2.39e23T^{2} \) |
| 5 | \( 1 - 1.83e17T + 1.77e34T^{2} \) |
| 7 | \( 1 + 6.94e20T + 2.56e41T^{2} \) |
| 11 | \( 1 + 3.82e25T + 1.06e51T^{2} \) |
| 13 | \( 1 + 2.81e27T + 3.83e54T^{2} \) |
| 17 | \( 1 + 8.61e28T + 1.95e60T^{2} \) |
| 19 | \( 1 - 2.22e31T + 4.55e62T^{2} \) |
| 23 | \( 1 + 3.43e33T + 5.30e66T^{2} \) |
| 29 | \( 1 + 8.08e34T + 4.54e71T^{2} \) |
| 31 | \( 1 + 2.10e36T + 1.19e73T^{2} \) |
| 37 | \( 1 - 1.93e37T + 6.94e76T^{2} \) |
| 41 | \( 1 - 5.03e39T + 1.06e79T^{2} \) |
| 43 | \( 1 - 1.24e40T + 1.09e80T^{2} \) |
| 47 | \( 1 + 8.45e40T + 8.56e81T^{2} \) |
| 53 | \( 1 + 2.42e42T + 3.08e84T^{2} \) |
| 59 | \( 1 + 1.80e43T + 5.91e86T^{2} \) |
| 61 | \( 1 - 9.36e43T + 3.02e87T^{2} \) |
| 67 | \( 1 + 2.03e44T + 3.00e89T^{2} \) |
| 71 | \( 1 - 4.45e44T + 5.14e90T^{2} \) |
| 73 | \( 1 + 7.35e45T + 2.00e91T^{2} \) |
| 79 | \( 1 + 3.49e46T + 9.63e92T^{2} \) |
| 83 | \( 1 - 1.11e47T + 1.08e94T^{2} \) |
| 89 | \( 1 - 1.64e47T + 3.31e95T^{2} \) |
| 97 | \( 1 + 3.51e48T + 2.24e97T^{2} \) |
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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
−16.07009390458362862232076917801, −14.21704650252904920939280029947, −12.88315956060479463436269766843, −9.966871464894000697832937261643, −9.413814330920998732272832278168, −7.58890240033698295300761005100, −5.80957635182560215543708673587, −2.93329524727049682992831078618, −2.16510977081162991618350910538, 0,
2.16510977081162991618350910538, 2.93329524727049682992831078618, 5.80957635182560215543708673587, 7.58890240033698295300761005100, 9.413814330920998732272832278168, 9.966871464894000697832937261643, 12.88315956060479463436269766843, 14.21704650252904920939280029947, 16.07009390458362862232076917801
