L(s) = 1 | + 3.67e11·2-s − 8.83e17·3-s + 9.72e22·4-s + 2.43e26·5-s − 3.24e29·6-s + 1.21e31·7-s + 2.18e34·8-s + 1.71e35·9-s + 8.95e37·10-s − 5.65e38·11-s − 8.58e40·12-s + 5.30e40·13-s + 4.46e42·14-s − 2.15e44·15-s + 4.35e45·16-s + 1.14e45·17-s + 6.30e46·18-s + 1.24e48·19-s + 2.36e49·20-s − 1.07e49·21-s − 2.07e50·22-s + 8.02e50·23-s − 1.92e52·24-s + 3.28e52·25-s + 1.94e52·26-s + 3.85e53·27-s + 1.18e54·28-s + ⋯ |
L(s) = 1 | + 1.89·2-s − 1.13·3-s + 2.57·4-s + 1.49·5-s − 2.14·6-s + 0.247·7-s + 2.97·8-s + 0.281·9-s + 2.83·10-s − 0.501·11-s − 2.91·12-s + 0.0894·13-s + 0.467·14-s − 1.69·15-s + 3.05·16-s + 0.0828·17-s + 0.533·18-s + 1.38·19-s + 3.85·20-s − 0.280·21-s − 0.947·22-s + 0.691·23-s − 3.36·24-s + 1.24·25-s + 0.169·26-s + 0.812·27-s + 0.636·28-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
\(L(38)\) |
\(\approx\) |
\(6.294403841\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.294403841\) |
\(L(\frac{77}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 - 3.67e11T + 3.77e22T^{2} \) |
| 3 | \( 1 + 8.83e17T + 6.08e35T^{2} \) |
| 5 | \( 1 - 2.43e26T + 2.64e52T^{2} \) |
| 7 | \( 1 - 1.21e31T + 2.41e63T^{2} \) |
| 11 | \( 1 + 5.65e38T + 1.27e78T^{2} \) |
| 13 | \( 1 - 5.30e40T + 3.51e83T^{2} \) |
| 17 | \( 1 - 1.14e45T + 1.92e92T^{2} \) |
| 19 | \( 1 - 1.24e48T + 8.06e95T^{2} \) |
| 23 | \( 1 - 8.02e50T + 1.34e102T^{2} \) |
| 29 | \( 1 - 8.53e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 6.18e55T + 7.11e111T^{2} \) |
| 37 | \( 1 + 5.98e58T + 4.12e117T^{2} \) |
| 41 | \( 1 + 5.32e60T + 9.09e120T^{2} \) |
| 43 | \( 1 - 2.59e61T + 3.23e122T^{2} \) |
| 47 | \( 1 + 5.76e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 5.92e63T + 2.09e129T^{2} \) |
| 59 | \( 1 - 2.82e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 5.43e66T + 7.93e133T^{2} \) |
| 67 | \( 1 + 2.61e68T + 9.02e136T^{2} \) |
| 71 | \( 1 - 1.78e67T + 6.98e138T^{2} \) |
| 73 | \( 1 + 1.09e70T + 5.61e139T^{2} \) |
| 79 | \( 1 + 1.49e71T + 2.09e142T^{2} \) |
| 83 | \( 1 - 3.39e71T + 8.52e143T^{2} \) |
| 89 | \( 1 - 2.45e72T + 1.60e146T^{2} \) |
| 97 | \( 1 - 1.36e74T + 1.01e149T^{2} \) |
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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
−16.16229169962515869019078375745, −14.27440203825751164593063510913, −13.15523880421666537234992680990, −11.75483711706298129143441610081, −10.42360721707807538206410562582, −6.74678126540167852088474308539, −5.58577562851286952894089240118, −5.03437170072694803087955500738, −2.91172135971785883778611554615, −1.45137740352345554509119398267,
1.45137740352345554509119398267, 2.91172135971785883778611554615, 5.03437170072694803087955500738, 5.58577562851286952894089240118, 6.74678126540167852088474308539, 10.42360721707807538206410562582, 11.75483711706298129143441610081, 13.15523880421666537234992680990, 14.27440203825751164593063510913, 16.16229169962515869019078375745