| L(s) = 1 | + 2.54e9·2-s − 9.84e14·3-s − 2.75e18·4-s − 1.69e22·5-s − 2.50e24·6-s + 1.59e26·7-s − 3.04e28·8-s − 1.75e29·9-s − 4.30e31·10-s + 8.86e32·11-s + 2.70e33·12-s + 7.82e34·13-s + 4.05e35·14-s + 1.66e37·15-s − 5.21e37·16-s − 5.03e38·17-s − 4.45e38·18-s + 8.03e39·19-s + 4.65e40·20-s − 1.56e41·21-s + 2.25e42·22-s − 1.14e43·23-s + 2.99e43·24-s + 1.77e44·25-s + 1.99e44·26-s + 1.29e45·27-s − 4.38e44·28-s + ⋯ |
| L(s) = 1 | + 0.837·2-s − 0.920·3-s − 0.298·4-s − 1.62·5-s − 0.770·6-s + 0.381·7-s − 1.08·8-s − 0.153·9-s − 1.36·10-s + 1.39·11-s + 0.274·12-s + 0.637·13-s + 0.319·14-s + 1.49·15-s − 0.612·16-s − 0.876·17-s − 0.128·18-s + 0.420·19-s + 0.484·20-s − 0.351·21-s + 1.16·22-s − 1.46·23-s + 1.00·24-s + 1.63·25-s + 0.533·26-s + 1.06·27-s − 0.113·28-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
| \(L(32)\) |
\(\approx\) |
\(1.013537659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013537659\) |
| \(L(\frac{65}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.54e9T + 9.22e18T^{2} \) |
| 3 | \( 1 + 9.84e14T + 1.14e30T^{2} \) |
| 5 | \( 1 + 1.69e22T + 1.08e44T^{2} \) |
| 7 | \( 1 - 1.59e26T + 1.74e53T^{2} \) |
| 11 | \( 1 - 8.86e32T + 4.05e65T^{2} \) |
| 13 | \( 1 - 7.82e34T + 1.50e70T^{2} \) |
| 17 | \( 1 + 5.03e38T + 3.29e77T^{2} \) |
| 19 | \( 1 - 8.03e39T + 3.64e80T^{2} \) |
| 23 | \( 1 + 1.14e43T + 6.14e85T^{2} \) |
| 29 | \( 1 - 2.15e46T + 1.35e92T^{2} \) |
| 31 | \( 1 + 6.36e46T + 9.03e93T^{2} \) |
| 37 | \( 1 + 1.05e49T + 6.26e98T^{2} \) |
| 41 | \( 1 - 8.54e50T + 4.03e101T^{2} \) |
| 43 | \( 1 + 3.25e51T + 8.10e102T^{2} \) |
| 47 | \( 1 - 1.11e52T + 2.19e105T^{2} \) |
| 53 | \( 1 + 5.90e53T + 4.25e108T^{2} \) |
| 59 | \( 1 - 2.16e55T + 3.66e111T^{2} \) |
| 61 | \( 1 - 1.72e56T + 2.99e112T^{2} \) |
| 67 | \( 1 + 2.22e57T + 1.10e115T^{2} \) |
| 71 | \( 1 - 2.30e58T + 4.25e116T^{2} \) |
| 73 | \( 1 - 1.23e58T + 2.45e117T^{2} \) |
| 79 | \( 1 - 8.71e58T + 3.55e119T^{2} \) |
| 83 | \( 1 + 1.10e60T + 7.97e120T^{2} \) |
| 89 | \( 1 - 4.52e61T + 6.47e122T^{2} \) |
| 97 | \( 1 - 2.14e62T + 1.46e125T^{2} \) |
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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
−17.84510587148979314760771788611, −15.94357929245443565257933538376, −14.31955506474695380971417929363, −12.13083194266712767185974727948, −11.40579155133014556535306382769, −8.521125339432963339889268140313, −6.36932168417662856012186571069, −4.64415932938404842791853805292, −3.65374290931023038947315699063, −0.60824199369184346752501954201,
0.60824199369184346752501954201, 3.65374290931023038947315699063, 4.64415932938404842791853805292, 6.36932168417662856012186571069, 8.521125339432963339889268140313, 11.40579155133014556535306382769, 12.13083194266712767185974727948, 14.31955506474695380971417929363, 15.94357929245443565257933538376, 17.84510587148979314760771788611
