| L(s) = 1 | − 2.05e10·2-s + 5.00e16·3-s − 1.93e21·4-s − 6.43e24·5-s − 1.02e27·6-s + 4.31e29·7-s + 8.84e31·8-s + 2.50e33·9-s + 1.32e35·10-s − 1.03e37·11-s − 9.69e37·12-s − 1.12e39·13-s − 8.87e39·14-s − 3.22e41·15-s + 2.75e42·16-s + 2.79e43·17-s − 5.14e43·18-s + 1.59e45·19-s + 1.24e46·20-s + 2.16e46·21-s + 2.12e47·22-s + 1.17e48·23-s + 4.42e48·24-s − 9.21e47·25-s + 2.31e49·26-s + 1.25e50·27-s − 8.36e50·28-s + ⋯ |
| L(s) = 1 | − 0.423·2-s + 0.577·3-s − 0.820·4-s − 0.989·5-s − 0.244·6-s + 0.430·7-s + 0.770·8-s + 0.333·9-s + 0.418·10-s − 1.11·11-s − 0.473·12-s − 0.320·13-s − 0.182·14-s − 0.571·15-s + 0.494·16-s + 0.583·17-s − 0.141·18-s + 0.642·19-s + 0.811·20-s + 0.248·21-s + 0.469·22-s + 0.536·23-s + 0.444·24-s − 0.0217·25-s + 0.135·26-s + 0.192·27-s − 0.353·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+71/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(36)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.00e16T \) |
| good | 2 | \( 1 + 2.05e10T + 2.36e21T^{2} \) |
| 5 | \( 1 + 6.43e24T + 4.23e49T^{2} \) |
| 7 | \( 1 - 4.31e29T + 1.00e60T^{2} \) |
| 11 | \( 1 + 1.03e37T + 8.68e73T^{2} \) |
| 13 | \( 1 + 1.12e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 2.79e43T + 2.30e87T^{2} \) |
| 19 | \( 1 - 1.59e45T + 6.18e90T^{2} \) |
| 23 | \( 1 - 1.17e48T + 4.81e96T^{2} \) |
| 29 | \( 1 - 1.13e52T + 6.76e103T^{2} \) |
| 31 | \( 1 + 8.82e52T + 7.70e105T^{2} \) |
| 37 | \( 1 - 5.62e55T + 2.19e111T^{2} \) |
| 41 | \( 1 + 1.33e56T + 3.21e114T^{2} \) |
| 43 | \( 1 - 1.13e58T + 9.46e115T^{2} \) |
| 47 | \( 1 - 1.12e59T + 5.23e118T^{2} \) |
| 53 | \( 1 - 8.51e60T + 2.65e122T^{2} \) |
| 59 | \( 1 + 1.21e62T + 5.37e125T^{2} \) |
| 61 | \( 1 + 4.10e63T + 5.73e126T^{2} \) |
| 67 | \( 1 + 5.67e64T + 4.48e129T^{2} \) |
| 71 | \( 1 + 1.78e65T + 2.75e131T^{2} \) |
| 73 | \( 1 - 1.82e66T + 1.97e132T^{2} \) |
| 79 | \( 1 - 3.72e67T + 5.38e134T^{2} \) |
| 83 | \( 1 + 2.02e68T + 1.79e136T^{2} \) |
| 89 | \( 1 - 2.01e69T + 2.55e138T^{2} \) |
| 97 | \( 1 + 4.57e70T + 1.15e141T^{2} \) |
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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
−12.37443226339481154138989391172, −10.72598503031419120714560280607, −9.405413560555663423307375829333, −8.091192972815441711161947003178, −7.56863702071020436651264099025, −5.18295352248873449622415152215, −4.12126180136199491769560562534, −2.84239566248726993243063805310, −1.13028142172654292726558837275, 0,
1.13028142172654292726558837275, 2.84239566248726993243063805310, 4.12126180136199491769560562534, 5.18295352248873449622415152215, 7.56863702071020436651264099025, 8.091192972815441711161947003178, 9.405413560555663423307375829333, 10.72598503031419120714560280607, 12.37443226339481154138989391172
