| L(s) = 1 | + 2.09e6·2-s + 2.36e10·3-s + 4.39e12·4-s − 1.55e15·5-s + 4.95e16·6-s + 5.58e17·7-s + 9.22e18·8-s + 2.29e20·9-s − 3.26e21·10-s − 5.19e19·11-s + 1.03e23·12-s − 9.74e23·13-s + 1.17e24·14-s − 3.67e25·15-s + 1.93e25·16-s + 4.85e26·17-s + 4.81e26·18-s − 3.40e27·19-s − 6.83e27·20-s + 1.31e28·21-s − 1.08e26·22-s + 2.45e29·23-s + 2.17e29·24-s + 1.28e30·25-s − 2.04e30·26-s − 2.33e30·27-s + 2.45e30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.30·3-s + 0.5·4-s − 1.45·5-s + 0.921·6-s + 0.377·7-s + 0.353·8-s + 0.698·9-s − 1.03·10-s − 0.00211·11-s + 0.651·12-s − 1.09·13-s + 0.267·14-s − 1.90·15-s + 0.250·16-s + 1.70·17-s + 0.494·18-s − 1.09·19-s − 0.729·20-s + 0.492·21-s − 0.00149·22-s + 1.29·23-s + 0.460·24-s + 1.12·25-s − 0.773·26-s − 0.392·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(\approx\) |
\(4.700701618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.700701618\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.09e6T \) |
| 7 | \( 1 - 5.58e17T \) |
| good | 3 | \( 1 - 2.36e10T + 3.28e20T^{2} \) |
| 5 | \( 1 + 1.55e15T + 1.13e30T^{2} \) |
| 11 | \( 1 + 5.19e19T + 6.02e44T^{2} \) |
| 13 | \( 1 + 9.74e23T + 7.93e47T^{2} \) |
| 17 | \( 1 - 4.85e26T + 8.11e52T^{2} \) |
| 19 | \( 1 + 3.40e27T + 9.69e54T^{2} \) |
| 23 | \( 1 - 2.45e29T + 3.58e58T^{2} \) |
| 29 | \( 1 - 5.23e31T + 7.64e62T^{2} \) |
| 31 | \( 1 + 9.91e31T + 1.34e64T^{2} \) |
| 37 | \( 1 + 6.08e33T + 2.70e67T^{2} \) |
| 41 | \( 1 + 2.43e34T + 2.23e69T^{2} \) |
| 43 | \( 1 - 2.00e35T + 1.73e70T^{2} \) |
| 47 | \( 1 - 7.63e35T + 7.94e71T^{2} \) |
| 53 | \( 1 - 1.58e37T + 1.39e74T^{2} \) |
| 59 | \( 1 - 1.04e38T + 1.40e76T^{2} \) |
| 61 | \( 1 + 2.80e37T + 5.87e76T^{2} \) |
| 67 | \( 1 - 2.92e39T + 3.32e78T^{2} \) |
| 71 | \( 1 + 1.92e39T + 4.01e79T^{2} \) |
| 73 | \( 1 + 6.17e39T + 1.32e80T^{2} \) |
| 79 | \( 1 - 8.86e40T + 3.96e81T^{2} \) |
| 83 | \( 1 + 7.53e40T + 3.31e82T^{2} \) |
| 89 | \( 1 - 1.03e42T + 6.66e83T^{2} \) |
| 97 | \( 1 - 3.33e41T + 2.69e85T^{2} \) |
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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
−11.84795900478120553328359844236, −10.42828454973305337938527018401, −8.779557208915709982011783706358, −7.85050329254615865660258491254, −7.11934350542388044171742147379, −5.13545107531820974154537102354, −4.04140624501641521775828594447, −3.22809303891151930131739215122, −2.32362907757557068625828709907, −0.792331854165344211758359807651,
0.792331854165344211758359807651, 2.32362907757557068625828709907, 3.22809303891151930131739215122, 4.04140624501641521775828594447, 5.13545107531820974154537102354, 7.11934350542388044171742147379, 7.85050329254615865660258491254, 8.779557208915709982011783706358, 10.42828454973305337938527018401, 11.84795900478120553328359844236
