| L(s) = 1 | − 1.37e11·2-s − 3.89e17·3-s + 1.88e22·4-s + 7.75e25·5-s + 5.35e28·6-s − 4.72e31·7-s − 2.59e33·8-s − 4.56e35·9-s − 1.06e37·10-s − 5.11e38·11-s − 7.36e39·12-s + 4.50e41·13-s + 6.49e42·14-s − 3.02e43·15-s + 3.56e44·16-s + 5.70e45·17-s + 6.27e46·18-s − 3.51e47·19-s + 1.46e48·20-s + 1.84e49·21-s + 7.03e49·22-s − 1.50e51·23-s + 1.01e51·24-s − 2.04e52·25-s − 6.19e52·26-s + 4.14e53·27-s − 8.93e53·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.499·3-s + 0.5·4-s + 0.476·5-s + 0.353·6-s − 0.962·7-s − 0.353·8-s − 0.750·9-s − 0.337·10-s − 0.453·11-s − 0.249·12-s + 0.760·13-s + 0.680·14-s − 0.238·15-s + 0.250·16-s + 0.411·17-s + 0.530·18-s − 0.391·19-s + 0.238·20-s + 0.481·21-s + 0.320·22-s − 1.29·23-s + 0.176·24-s − 0.772·25-s − 0.537·26-s + 0.874·27-s − 0.481·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(0.6131251899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6131251899\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.37e11T \) |
| good | 3 | \( 1 + 3.89e17T + 6.08e35T^{2} \) |
| 5 | \( 1 - 7.75e25T + 2.64e52T^{2} \) |
| 7 | \( 1 + 4.72e31T + 2.41e63T^{2} \) |
| 11 | \( 1 + 5.11e38T + 1.27e78T^{2} \) |
| 13 | \( 1 - 4.50e41T + 3.51e83T^{2} \) |
| 17 | \( 1 - 5.70e45T + 1.92e92T^{2} \) |
| 19 | \( 1 + 3.51e47T + 8.06e95T^{2} \) |
| 23 | \( 1 + 1.50e51T + 1.34e102T^{2} \) |
| 29 | \( 1 + 4.96e54T + 4.78e109T^{2} \) |
| 31 | \( 1 - 4.96e55T + 7.11e111T^{2} \) |
| 37 | \( 1 + 9.67e58T + 4.12e117T^{2} \) |
| 41 | \( 1 + 1.27e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 2.20e60T + 3.23e122T^{2} \) |
| 47 | \( 1 - 4.95e62T + 2.55e125T^{2} \) |
| 53 | \( 1 + 4.82e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 1.51e65T + 6.51e132T^{2} \) |
| 61 | \( 1 - 6.05e66T + 7.93e133T^{2} \) |
| 67 | \( 1 + 2.23e68T + 9.02e136T^{2} \) |
| 71 | \( 1 - 2.32e69T + 6.98e138T^{2} \) |
| 73 | \( 1 - 1.23e70T + 5.61e139T^{2} \) |
| 79 | \( 1 - 2.33e71T + 2.09e142T^{2} \) |
| 83 | \( 1 - 1.73e72T + 8.52e143T^{2} \) |
| 89 | \( 1 + 1.75e73T + 1.60e146T^{2} \) |
| 97 | \( 1 + 4.85e74T + 1.01e149T^{2} \) |
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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
−13.76458232583470012093177878212, −12.19532409584841626743876842387, −10.74931622088717196271561565925, −9.615820511184229326661561775550, −8.214874778419337713668843499059, −6.47805313487667273496264150563, −5.61338238573709102569140064268, −3.43532005742058874247842214448, −2.02843695096353448089937647280, −0.44360186992874782788816496510,
0.44360186992874782788816496510, 2.02843695096353448089937647280, 3.43532005742058874247842214448, 5.61338238573709102569140064268, 6.47805313487667273496264150563, 8.214874778419337713668843499059, 9.615820511184229326661561775550, 10.74931622088717196271561565925, 12.19532409584841626743876842387, 13.76458232583470012093177878212
