| L(s) = 1 | − 5.49e11·2-s + 1.08e18·3-s + 3.02e23·4-s + 4.57e27·5-s − 5.95e29·6-s + 3.27e33·7-s − 1.66e35·8-s − 4.80e37·9-s − 2.51e39·10-s + 1.84e41·11-s + 3.27e41·12-s + 1.55e44·13-s − 1.80e45·14-s + 4.96e45·15-s + 9.13e46·16-s + 6.83e48·17-s + 2.64e49·18-s − 2.17e50·19-s + 1.38e51·20-s + 3.55e51·21-s − 1.01e53·22-s + 9.98e53·23-s − 1.80e53·24-s + 4.42e54·25-s − 8.55e55·26-s − 1.05e56·27-s + 9.91e56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.154·3-s + 0.5·4-s + 1.12·5-s − 0.109·6-s + 1.36·7-s − 0.353·8-s − 0.976·9-s − 0.796·10-s + 1.35·11-s + 0.0771·12-s + 1.55·13-s − 0.963·14-s + 0.173·15-s + 0.250·16-s + 1.70·17-s + 0.690·18-s − 0.670·19-s + 0.562·20-s + 0.210·21-s − 0.955·22-s + 1.62·23-s − 0.0545·24-s + 0.267·25-s − 1.09·26-s − 0.305·27-s + 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(80-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+79/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(40)\) |
\(\approx\) |
\(3.012884808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.012884808\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.49e11T \) |
| good | 3 | \( 1 - 1.08e18T + 4.92e37T^{2} \) |
| 5 | \( 1 - 4.57e27T + 1.65e55T^{2} \) |
| 7 | \( 1 - 3.27e33T + 5.79e66T^{2} \) |
| 11 | \( 1 - 1.84e41T + 1.86e82T^{2} \) |
| 13 | \( 1 - 1.55e44T + 1.00e88T^{2} \) |
| 17 | \( 1 - 6.83e48T + 1.60e97T^{2} \) |
| 19 | \( 1 + 2.17e50T + 1.05e101T^{2} \) |
| 23 | \( 1 - 9.98e53T + 3.77e107T^{2} \) |
| 29 | \( 1 - 4.67e57T + 3.38e115T^{2} \) |
| 31 | \( 1 + 1.58e58T + 6.57e117T^{2} \) |
| 37 | \( 1 + 1.16e62T + 7.72e123T^{2} \) |
| 41 | \( 1 + 5.97e63T + 2.56e127T^{2} \) |
| 43 | \( 1 + 2.46e64T + 1.10e129T^{2} \) |
| 47 | \( 1 - 3.19e65T + 1.24e132T^{2} \) |
| 53 | \( 1 + 1.08e68T + 1.65e136T^{2} \) |
| 59 | \( 1 + 1.11e70T + 7.89e139T^{2} \) |
| 61 | \( 1 - 3.15e70T + 1.09e141T^{2} \) |
| 67 | \( 1 + 3.92e71T + 1.81e144T^{2} \) |
| 71 | \( 1 - 3.07e72T + 1.77e146T^{2} \) |
| 73 | \( 1 + 3.14e73T + 1.59e147T^{2} \) |
| 79 | \( 1 - 9.72e74T + 8.17e149T^{2} \) |
| 83 | \( 1 + 2.59e75T + 4.04e151T^{2} \) |
| 89 | \( 1 - 1.14e77T + 1.00e154T^{2} \) |
| 97 | \( 1 - 3.06e78T + 9.01e156T^{2} \) |
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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
−13.86897616517943210213236539669, −11.75958605600826959229716112501, −10.65619951151893860352900693847, −9.080851012486346389202907741669, −8.276874387095074604788700531755, −6.45378977927907945339973267337, −5.32506121140665397845331476727, −3.32489859161175420727318797625, −1.68765511688187830613714208588, −1.13988817728963692237682202629,
1.13988817728963692237682202629, 1.68765511688187830613714208588, 3.32489859161175420727318797625, 5.32506121140665397845331476727, 6.45378977927907945339973267337, 8.276874387095074604788700531755, 9.080851012486346389202907741669, 10.65619951151893860352900693847, 11.75958605600826959229716112501, 13.86897616517943210213236539669
