| L(s) = 1 | + 1.09e12·2-s − 4.08e19·3-s + 1.20e24·4-s + 2.59e28·5-s − 4.48e31·6-s + 1.64e34·7-s + 1.32e36·8-s + 1.22e39·9-s + 2.85e40·10-s + 2.10e42·11-s − 4.93e43·12-s + 6.02e44·13-s + 1.81e46·14-s − 1.05e48·15-s + 1.46e48·16-s + 6.31e49·17-s + 1.34e51·18-s + 2.57e51·19-s + 3.13e52·20-s − 6.72e53·21-s + 2.31e54·22-s + 8.80e54·23-s − 5.42e55·24-s + 2.60e56·25-s + 6.62e56·26-s − 3.17e58·27-s + 1.99e58·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.93·3-s + 0.5·4-s + 1.27·5-s − 1.37·6-s + 0.977·7-s + 0.353·8-s + 2.75·9-s + 0.902·10-s + 1.40·11-s − 0.969·12-s + 0.462·13-s + 0.691·14-s − 2.47·15-s + 0.250·16-s + 0.927·17-s + 1.94·18-s + 0.417·19-s + 0.638·20-s − 1.89·21-s + 0.991·22-s + 0.623·23-s − 0.685·24-s + 0.628·25-s + 0.327·26-s − 3.40·27-s + 0.488·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(82-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+81/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(41)\) |
\(\approx\) |
\(3.430038071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.430038071\) |
| \(L(\frac{83}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.09e12T \) |
| good | 3 | \( 1 + 4.08e19T + 4.43e38T^{2} \) |
| 5 | \( 1 - 2.59e28T + 4.13e56T^{2} \) |
| 7 | \( 1 - 1.64e34T + 2.83e68T^{2} \) |
| 11 | \( 1 - 2.10e42T + 2.25e84T^{2} \) |
| 13 | \( 1 - 6.02e44T + 1.69e90T^{2} \) |
| 17 | \( 1 - 6.31e49T + 4.63e99T^{2} \) |
| 19 | \( 1 - 2.57e51T + 3.79e103T^{2} \) |
| 23 | \( 1 - 8.80e54T + 1.99e110T^{2} \) |
| 29 | \( 1 + 2.59e59T + 2.84e118T^{2} \) |
| 31 | \( 1 + 7.51e59T + 6.31e120T^{2} \) |
| 37 | \( 1 - 3.27e63T + 1.05e127T^{2} \) |
| 41 | \( 1 - 5.29e64T + 4.32e130T^{2} \) |
| 43 | \( 1 + 7.79e65T + 2.04e132T^{2} \) |
| 47 | \( 1 - 1.36e67T + 2.75e135T^{2} \) |
| 53 | \( 1 - 9.04e69T + 4.63e139T^{2} \) |
| 59 | \( 1 - 2.63e71T + 2.74e143T^{2} \) |
| 61 | \( 1 - 2.00e72T + 4.08e144T^{2} \) |
| 67 | \( 1 + 1.04e74T + 8.16e147T^{2} \) |
| 71 | \( 1 + 6.00e73T + 8.95e149T^{2} \) |
| 73 | \( 1 - 1.61e75T + 8.49e150T^{2} \) |
| 79 | \( 1 + 7.95e76T + 5.10e153T^{2} \) |
| 83 | \( 1 + 8.38e77T + 2.78e155T^{2} \) |
| 89 | \( 1 + 1.88e78T + 7.95e157T^{2} \) |
| 97 | \( 1 + 3.96e80T + 8.48e160T^{2} \) |
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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
−13.18786254457292754268156812743, −11.84679657610642598384438310409, −11.04343429402192610029217130850, −9.703172740727892443007957072052, −7.06172054261726660448778580391, −5.90454407952346430133305049148, −5.35298506312240518733494336163, −4.09308719670146413002772945969, −1.66027834677990205904022508826, −1.07905802816476386157795905783,
1.07905802816476386157795905783, 1.66027834677990205904022508826, 4.09308719670146413002772945969, 5.35298506312240518733494336163, 5.90454407952346430133305049148, 7.06172054261726660448778580391, 9.703172740727892443007957072052, 11.04343429402192610029217130850, 11.84679657610642598384438310409, 13.18786254457292754268156812743
