| L(s) = 1 | − 4.68e9·2-s − 3.45e14·3-s + 1.27e19·4-s − 3.68e21·5-s + 1.61e24·6-s + 6.76e26·7-s − 1.63e28·8-s − 1.02e30·9-s + 1.72e31·10-s − 4.60e32·11-s − 4.39e33·12-s − 1.39e35·13-s − 3.17e36·14-s + 1.27e36·15-s − 4.07e37·16-s + 6.98e38·17-s + 4.80e39·18-s − 2.26e40·19-s − 4.68e40·20-s − 2.34e41·21-s + 2.15e42·22-s + 9.79e42·23-s + 5.65e42·24-s − 9.48e43·25-s + 6.53e44·26-s + 7.50e44·27-s + 8.60e45·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 0.323·3-s + 1.37·4-s − 0.353·5-s + 0.498·6-s + 1.62·7-s − 0.583·8-s − 0.895·9-s + 0.545·10-s − 0.723·11-s − 0.445·12-s − 1.13·13-s − 2.50·14-s + 0.114·15-s − 0.478·16-s + 1.21·17-s + 1.38·18-s − 1.18·19-s − 0.487·20-s − 0.524·21-s + 1.11·22-s + 1.24·23-s + 0.188·24-s − 0.874·25-s + 1.75·26-s + 0.612·27-s + 2.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(64-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+63/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(32)\) |
\(\approx\) |
\(0.6102351277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6102351277\) |
| \(L(\frac{65}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 4.68e9T + 9.22e18T^{2} \) |
| 3 | \( 1 + 3.45e14T + 1.14e30T^{2} \) |
| 5 | \( 1 + 3.68e21T + 1.08e44T^{2} \) |
| 7 | \( 1 - 6.76e26T + 1.74e53T^{2} \) |
| 11 | \( 1 + 4.60e32T + 4.05e65T^{2} \) |
| 13 | \( 1 + 1.39e35T + 1.50e70T^{2} \) |
| 17 | \( 1 - 6.98e38T + 3.29e77T^{2} \) |
| 19 | \( 1 + 2.26e40T + 3.64e80T^{2} \) |
| 23 | \( 1 - 9.79e42T + 6.14e85T^{2} \) |
| 29 | \( 1 - 1.02e46T + 1.35e92T^{2} \) |
| 31 | \( 1 + 7.88e46T + 9.03e93T^{2} \) |
| 37 | \( 1 + 4.47e48T + 6.26e98T^{2} \) |
| 41 | \( 1 + 2.61e50T + 4.03e101T^{2} \) |
| 43 | \( 1 - 1.36e51T + 8.10e102T^{2} \) |
| 47 | \( 1 + 1.83e52T + 2.19e105T^{2} \) |
| 53 | \( 1 - 3.07e54T + 4.25e108T^{2} \) |
| 59 | \( 1 - 6.07e55T + 3.66e111T^{2} \) |
| 61 | \( 1 - 2.59e56T + 2.99e112T^{2} \) |
| 67 | \( 1 + 2.81e57T + 1.10e115T^{2} \) |
| 71 | \( 1 + 2.35e58T + 4.25e116T^{2} \) |
| 73 | \( 1 - 3.66e58T + 2.45e117T^{2} \) |
| 79 | \( 1 - 9.89e59T + 3.55e119T^{2} \) |
| 83 | \( 1 - 3.72e59T + 7.97e120T^{2} \) |
| 89 | \( 1 - 2.65e61T + 6.47e122T^{2} \) |
| 97 | \( 1 - 2.06e62T + 1.46e125T^{2} \) |
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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
−17.79506601430641543509648962802, −16.86783836802395123605928443560, −14.75113690352835432131230345710, −11.68596911820508741304314952304, −10.49602289767242460507648530691, −8.528091325463947058665729713447, −7.54164017761432726094566132719, −5.07839043795559135938970624058, −2.21468931928119782116171883813, −0.64534129392468597418821265851,
0.64534129392468597418821265851, 2.21468931928119782116171883813, 5.07839043795559135938970624058, 7.54164017761432726094566132719, 8.528091325463947058665729713447, 10.49602289767242460507648530691, 11.68596911820508741304314952304, 14.75113690352835432131230345710, 16.86783836802395123605928443560, 17.79506601430641543509648962802
