| L(s) = 1 | + 2.16e13·2-s − 4.57e21·3-s − 2.00e27·4-s + 4.97e31·5-s − 9.90e34·6-s − 2.82e38·7-s − 9.69e40·8-s − 5.21e42·9-s + 1.07e45·10-s + 1.15e47·11-s + 9.19e48·12-s − 6.90e50·13-s − 6.11e51·14-s − 2.27e53·15-s + 2.87e54·16-s − 5.85e55·17-s − 1.12e56·18-s − 2.50e58·19-s − 9.99e58·20-s + 1.29e60·21-s + 2.50e60·22-s − 1.24e60·23-s + 4.44e62·24-s − 1.56e63·25-s − 1.49e64·26-s + 1.43e65·27-s + 5.67e65·28-s + ⋯ |
| L(s) = 1 | + 0.434·2-s − 0.894·3-s − 0.811·4-s + 0.783·5-s − 0.388·6-s − 0.998·7-s − 0.787·8-s − 0.199·9-s + 0.340·10-s + 0.479·11-s + 0.725·12-s − 1.42·13-s − 0.433·14-s − 0.701·15-s + 0.469·16-s − 0.605·17-s − 0.0865·18-s − 1.64·19-s − 0.635·20-s + 0.893·21-s + 0.208·22-s − 0.0136·23-s + 0.704·24-s − 0.386·25-s − 0.620·26-s + 1.07·27-s + 0.809·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(92-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+91/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(46)\) |
\(\approx\) |
\(0.5801020588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5801020588\) |
| \(L(\frac{93}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.16e13T + 2.47e27T^{2} \) |
| 3 | \( 1 + 4.57e21T + 2.61e43T^{2} \) |
| 5 | \( 1 - 4.97e31T + 4.03e63T^{2} \) |
| 7 | \( 1 + 2.82e38T + 8.01e76T^{2} \) |
| 11 | \( 1 - 1.15e47T + 5.84e94T^{2} \) |
| 13 | \( 1 + 6.90e50T + 2.33e101T^{2} \) |
| 17 | \( 1 + 5.85e55T + 9.35e111T^{2} \) |
| 19 | \( 1 + 2.50e58T + 2.32e116T^{2} \) |
| 23 | \( 1 + 1.24e60T + 8.26e123T^{2} \) |
| 29 | \( 1 - 1.70e66T + 1.19e133T^{2} \) |
| 31 | \( 1 + 3.90e67T + 5.17e135T^{2} \) |
| 37 | \( 1 - 2.18e71T + 5.08e142T^{2} \) |
| 41 | \( 1 + 2.33e73T + 5.79e146T^{2} \) |
| 43 | \( 1 - 1.45e74T + 4.42e148T^{2} \) |
| 47 | \( 1 - 1.72e76T + 1.44e152T^{2} \) |
| 53 | \( 1 - 3.47e78T + 8.11e156T^{2} \) |
| 59 | \( 1 - 5.44e80T + 1.40e161T^{2} \) |
| 61 | \( 1 - 9.48e80T + 2.91e162T^{2} \) |
| 67 | \( 1 + 2.20e83T + 1.48e166T^{2} \) |
| 71 | \( 1 + 3.41e84T + 2.91e168T^{2} \) |
| 73 | \( 1 - 1.00e85T + 3.65e169T^{2} \) |
| 79 | \( 1 + 7.90e84T + 4.83e172T^{2} \) |
| 83 | \( 1 + 8.82e86T + 4.32e174T^{2} \) |
| 89 | \( 1 - 6.33e88T + 2.48e177T^{2} \) |
| 97 | \( 1 + 2.81e88T + 6.25e180T^{2} \) |
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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
−14.70154436370938708080540817796, −13.23929588660988224660154127304, −12.14395701339489374025107992878, −10.20307757852418910868735561213, −9.011297663054050862815261101416, −6.53285362170240883325332394000, −5.57166719896395994507810463274, −4.27014730643878452349185256423, −2.50661147552419451212008011432, −0.39461795258496705328255038271,
0.39461795258496705328255038271, 2.50661147552419451212008011432, 4.27014730643878452349185256423, 5.57166719896395994507810463274, 6.53285362170240883325332394000, 9.011297663054050862815261101416, 10.20307757852418910868735561213, 12.14395701339489374025107992878, 13.23929588660988224660154127304, 14.70154436370938708080540817796
