| L(s) = 1 | − 2.89e11·2-s + 1.03e18·3-s + 4.61e22·4-s − 2.47e26·5-s − 3.00e29·6-s − 6.83e31·7-s − 2.41e33·8-s + 4.64e35·9-s + 7.16e37·10-s − 1.33e39·11-s + 4.77e40·12-s + 2.92e41·13-s + 1.97e43·14-s − 2.56e44·15-s − 1.04e45·16-s + 4.46e44·17-s − 1.34e47·18-s − 8.32e47·19-s − 1.14e49·20-s − 7.07e49·21-s + 3.86e50·22-s + 1.13e51·23-s − 2.50e51·24-s + 3.47e52·25-s − 8.47e52·26-s − 1.48e53·27-s − 3.15e54·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.32·3-s + 1.22·4-s − 1.52·5-s − 1.97·6-s − 1.39·7-s − 0.328·8-s + 0.763·9-s + 2.26·10-s − 1.18·11-s + 1.62·12-s + 0.493·13-s + 2.07·14-s − 2.02·15-s − 0.730·16-s + 0.0321·17-s − 1.13·18-s − 0.926·19-s − 1.85·20-s − 1.84·21-s + 1.76·22-s + 0.976·23-s − 0.436·24-s + 1.31·25-s − 0.735·26-s − 0.313·27-s − 1.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(76-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+75/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(0.4817922437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4817922437\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.89e11T + 3.77e22T^{2} \) |
| 3 | \( 1 - 1.03e18T + 6.08e35T^{2} \) |
| 5 | \( 1 + 2.47e26T + 2.64e52T^{2} \) |
| 7 | \( 1 + 6.83e31T + 2.41e63T^{2} \) |
| 11 | \( 1 + 1.33e39T + 1.27e78T^{2} \) |
| 13 | \( 1 - 2.92e41T + 3.51e83T^{2} \) |
| 17 | \( 1 - 4.46e44T + 1.92e92T^{2} \) |
| 19 | \( 1 + 8.32e47T + 8.06e95T^{2} \) |
| 23 | \( 1 - 1.13e51T + 1.34e102T^{2} \) |
| 29 | \( 1 - 3.24e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 1.48e56T + 7.11e111T^{2} \) |
| 37 | \( 1 - 2.52e58T + 4.12e117T^{2} \) |
| 41 | \( 1 + 1.57e60T + 9.09e120T^{2} \) |
| 43 | \( 1 - 2.24e61T + 3.23e122T^{2} \) |
| 47 | \( 1 - 3.99e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 2.56e64T + 2.09e129T^{2} \) |
| 59 | \( 1 - 1.98e66T + 6.51e132T^{2} \) |
| 61 | \( 1 + 1.77e66T + 7.93e133T^{2} \) |
| 67 | \( 1 - 3.79e68T + 9.02e136T^{2} \) |
| 71 | \( 1 - 1.15e69T + 6.98e138T^{2} \) |
| 73 | \( 1 + 5.75e69T + 5.61e139T^{2} \) |
| 79 | \( 1 - 6.59e70T + 2.09e142T^{2} \) |
| 83 | \( 1 - 1.04e72T + 8.52e143T^{2} \) |
| 89 | \( 1 - 1.34e73T + 1.60e146T^{2} \) |
| 97 | \( 1 + 5.17e74T + 1.01e149T^{2} \) |
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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
−16.27600116982798116923691995254, −15.33686387649280416526202783598, −12.99433846952762269445629494873, −10.73943178191518419056077062310, −9.166434742618446545633211301906, −8.189914496586127309609984085416, −7.18004539728659984538861799288, −3.72886977494453316318830446873, −2.55408352758006985195857895610, −0.46629182134089608875322244810,
0.46629182134089608875322244810, 2.55408352758006985195857895610, 3.72886977494453316318830446873, 7.18004539728659984538861799288, 8.189914496586127309609984085416, 9.166434742618446545633211301906, 10.73943178191518419056077062310, 12.99433846952762269445629494873, 15.33686387649280416526202783598, 16.27600116982798116923691995254
