L(s) = 1 | + 2.30e7·2-s + 1.28e11·3-s + 3.91e14·4-s − 1.15e16·5-s + 2.96e18·6-s + 1.54e19·7-s + 5.79e21·8-s − 1.00e22·9-s − 2.65e23·10-s − 3.19e24·11-s + 5.03e25·12-s + 1.57e25·13-s + 3.55e26·14-s − 1.47e27·15-s + 7.86e28·16-s − 3.42e28·17-s − 2.32e29·18-s + 1.00e30·19-s − 4.51e30·20-s + 1.98e30·21-s − 7.37e31·22-s + 5.17e31·23-s + 7.44e32·24-s − 5.78e32·25-s + 3.64e32·26-s − 4.71e33·27-s + 6.04e33·28-s + ⋯ |
L(s) = 1 | + 1.94·2-s + 0.788·3-s + 2.78·4-s − 0.431·5-s + 1.53·6-s + 0.212·7-s + 3.47·8-s − 0.378·9-s − 0.839·10-s − 1.07·11-s + 2.19·12-s + 0.104·13-s + 0.414·14-s − 0.340·15-s + 3.97·16-s − 0.416·17-s − 0.737·18-s + 0.889·19-s − 1.20·20-s + 0.167·21-s − 2.09·22-s + 0.517·23-s + 2.73·24-s − 0.813·25-s + 0.203·26-s − 1.08·27-s + 0.593·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(48-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+47/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(24)\) |
\(\approx\) |
\(6.573291442\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.573291442\) |
\(L(\frac{49}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 - 2.30e7T + 1.40e14T^{2} \) |
| 3 | \( 1 - 1.28e11T + 2.65e22T^{2} \) |
| 5 | \( 1 + 1.15e16T + 7.10e32T^{2} \) |
| 7 | \( 1 - 1.54e19T + 5.24e39T^{2} \) |
| 11 | \( 1 + 3.19e24T + 8.81e48T^{2} \) |
| 13 | \( 1 - 1.57e25T + 2.26e52T^{2} \) |
| 17 | \( 1 + 3.42e28T + 6.77e57T^{2} \) |
| 19 | \( 1 - 1.00e30T + 1.26e60T^{2} \) |
| 23 | \( 1 - 5.17e31T + 1.00e64T^{2} \) |
| 29 | \( 1 + 6.90e33T + 5.40e68T^{2} \) |
| 31 | \( 1 + 1.79e35T + 1.24e70T^{2} \) |
| 37 | \( 1 + 2.68e36T + 5.07e73T^{2} \) |
| 41 | \( 1 - 1.17e38T + 6.32e75T^{2} \) |
| 43 | \( 1 - 1.58e38T + 5.92e76T^{2} \) |
| 47 | \( 1 + 1.34e39T + 3.87e78T^{2} \) |
| 53 | \( 1 - 4.68e40T + 1.09e81T^{2} \) |
| 59 | \( 1 - 3.84e41T + 1.69e83T^{2} \) |
| 61 | \( 1 - 4.37e41T + 8.13e83T^{2} \) |
| 67 | \( 1 - 4.64e42T + 6.69e85T^{2} \) |
| 71 | \( 1 - 1.91e43T + 1.02e87T^{2} \) |
| 73 | \( 1 - 8.25e43T + 3.76e87T^{2} \) |
| 79 | \( 1 + 5.45e44T + 1.54e89T^{2} \) |
| 83 | \( 1 + 1.02e45T + 1.57e90T^{2} \) |
| 89 | \( 1 - 1.06e46T + 4.18e91T^{2} \) |
| 97 | \( 1 - 3.72e46T + 2.38e93T^{2} \) |
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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
−21.07516266632649335123314739269, −19.88095368143506660305027662922, −15.82196180206301809420558655544, −14.48598525075133139211032382979, −13.11442815552733184083047136647, −11.28244498175268667193628020180, −7.59247495885116737306597296132, −5.40006152176154725533900714111, −3.64203140782896131177921063585, −2.34778698908042292893419555331,
2.34778698908042292893419555331, 3.64203140782896131177921063585, 5.40006152176154725533900714111, 7.59247495885116737306597296132, 11.28244498175268667193628020180, 13.11442815552733184083047136647, 14.48598525075133139211032382979, 15.82196180206301809420558655544, 19.88095368143506660305027662922, 21.07516266632649335123314739269