| L(s) = 1 | + 9.51e12·2-s + 5.28e21·3-s − 2.38e27·4-s − 1.05e32·5-s + 5.02e34·6-s − 1.59e38·7-s − 4.62e40·8-s + 1.77e42·9-s − 1.00e45·10-s − 4.48e47·11-s − 1.26e49·12-s + 5.36e49·13-s − 1.51e51·14-s − 5.58e53·15-s + 5.46e54·16-s + 1.62e56·17-s + 1.69e55·18-s + 6.01e57·19-s + 2.51e59·20-s − 8.43e59·21-s − 4.27e60·22-s − 6.17e61·23-s − 2.44e62·24-s + 7.10e63·25-s + 5.10e62·26-s − 1.29e65·27-s + 3.80e65·28-s + ⋯ |
| L(s) = 1 | + 0.191·2-s + 1.03·3-s − 0.963·4-s − 1.66·5-s + 0.197·6-s − 0.563·7-s − 0.375·8-s + 0.0679·9-s − 0.317·10-s − 1.85·11-s − 0.995·12-s + 0.110·13-s − 0.107·14-s − 1.71·15-s + 0.891·16-s + 1.67·17-s + 0.0129·18-s + 0.394·19-s + 1.60·20-s − 0.582·21-s − 0.355·22-s − 0.678·23-s − 0.387·24-s + 1.75·25-s + 0.0212·26-s − 0.963·27-s + 0.542·28-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
| \(L(46)\) |
\(\approx\) |
\(0.8477709179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8477709179\) |
| \(L(\frac{93}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 9.51e12T + 2.47e27T^{2} \) |
| 3 | \( 1 - 5.28e21T + 2.61e43T^{2} \) |
| 5 | \( 1 + 1.05e32T + 4.03e63T^{2} \) |
| 7 | \( 1 + 1.59e38T + 8.01e76T^{2} \) |
| 11 | \( 1 + 4.48e47T + 5.84e94T^{2} \) |
| 13 | \( 1 - 5.36e49T + 2.33e101T^{2} \) |
| 17 | \( 1 - 1.62e56T + 9.35e111T^{2} \) |
| 19 | \( 1 - 6.01e57T + 2.32e116T^{2} \) |
| 23 | \( 1 + 6.17e61T + 8.26e123T^{2} \) |
| 29 | \( 1 - 1.16e66T + 1.19e133T^{2} \) |
| 31 | \( 1 - 5.07e67T + 5.17e135T^{2} \) |
| 37 | \( 1 + 1.37e71T + 5.08e142T^{2} \) |
| 41 | \( 1 + 5.73e72T + 5.79e146T^{2} \) |
| 43 | \( 1 + 2.86e74T + 4.42e148T^{2} \) |
| 47 | \( 1 + 1.58e75T + 1.44e152T^{2} \) |
| 53 | \( 1 - 1.36e78T + 8.11e156T^{2} \) |
| 59 | \( 1 - 5.82e80T + 1.40e161T^{2} \) |
| 61 | \( 1 + 5.21e80T + 2.91e162T^{2} \) |
| 67 | \( 1 + 2.56e82T + 1.48e166T^{2} \) |
| 71 | \( 1 + 2.34e83T + 2.91e168T^{2} \) |
| 73 | \( 1 - 2.47e84T + 3.65e169T^{2} \) |
| 79 | \( 1 - 2.98e86T + 4.83e172T^{2} \) |
| 83 | \( 1 - 9.13e86T + 4.32e174T^{2} \) |
| 89 | \( 1 - 5.81e88T + 2.48e177T^{2} \) |
| 97 | \( 1 + 3.70e90T + 6.25e180T^{2} \) |
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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
−14.87640349981632150081121786618, −13.47704461884170816893900120387, −12.15652388170226721169470323253, −10.03142438430599959202702198550, −8.316011680211581958499960877557, −7.79353790360146699786275507370, −5.17913490787136325319949515808, −3.64985173005139083638954826698, −3.00603027898949892152837519263, −0.45655895695253198032077637165,
0.45655895695253198032077637165, 3.00603027898949892152837519263, 3.64985173005139083638954826698, 5.17913490787136325319949515808, 7.79353790360146699786275507370, 8.316011680211581958499960877557, 10.03142438430599959202702198550, 12.15652388170226721169470323253, 13.47704461884170816893900120387, 14.87640349981632150081121786618
