| L(s) = 1 | + 1.67e7·2-s + 3.20e11·3-s + 2.81e14·4-s − 1.37e17·5-s + 5.37e18·6-s − 1.91e20·7-s + 4.72e21·8-s − 1.36e23·9-s − 2.30e24·10-s + 1.44e25·11-s + 9.02e25·12-s − 1.12e27·13-s − 3.21e27·14-s − 4.40e28·15-s + 7.92e28·16-s − 1.48e30·17-s − 2.29e30·18-s + 3.08e31·19-s − 3.86e31·20-s − 6.14e31·21-s + 2.42e32·22-s − 1.52e33·23-s + 1.51e33·24-s + 1.09e33·25-s − 1.89e34·26-s − 1.20e35·27-s − 5.39e34·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.655·3-s + 0.5·4-s − 1.03·5-s + 0.463·6-s − 0.377·7-s + 0.353·8-s − 0.570·9-s − 0.728·10-s + 0.443·11-s + 0.327·12-s − 0.577·13-s − 0.267·14-s − 0.675·15-s + 0.250·16-s − 1.06·17-s − 0.403·18-s + 1.44·19-s − 0.515·20-s − 0.247·21-s + 0.313·22-s − 0.664·23-s + 0.231·24-s + 0.0614·25-s − 0.408·26-s − 1.02·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(25)\) |
\(\approx\) |
\(2.723834070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.723834070\) |
| \(L(\frac{51}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.67e7T \) |
| 7 | \( 1 + 1.91e20T \) |
| good | 3 | \( 1 - 3.20e11T + 2.39e23T^{2} \) |
| 5 | \( 1 + 1.37e17T + 1.77e34T^{2} \) |
| 11 | \( 1 - 1.44e25T + 1.06e51T^{2} \) |
| 13 | \( 1 + 1.12e27T + 3.83e54T^{2} \) |
| 17 | \( 1 + 1.48e30T + 1.95e60T^{2} \) |
| 19 | \( 1 - 3.08e31T + 4.55e62T^{2} \) |
| 23 | \( 1 + 1.52e33T + 5.30e66T^{2} \) |
| 29 | \( 1 - 9.41e34T + 4.54e71T^{2} \) |
| 31 | \( 1 - 5.15e36T + 1.19e73T^{2} \) |
| 37 | \( 1 + 2.16e38T + 6.94e76T^{2} \) |
| 41 | \( 1 + 2.53e39T + 1.06e79T^{2} \) |
| 43 | \( 1 - 1.22e40T + 1.09e80T^{2} \) |
| 47 | \( 1 + 1.81e41T + 8.56e81T^{2} \) |
| 53 | \( 1 - 4.05e41T + 3.08e84T^{2} \) |
| 59 | \( 1 - 4.58e43T + 5.91e86T^{2} \) |
| 61 | \( 1 - 2.26e43T + 3.02e87T^{2} \) |
| 67 | \( 1 - 2.24e44T + 3.00e89T^{2} \) |
| 71 | \( 1 + 4.01e44T + 5.14e90T^{2} \) |
| 73 | \( 1 - 2.86e44T + 2.00e91T^{2} \) |
| 79 | \( 1 - 5.06e46T + 9.63e92T^{2} \) |
| 83 | \( 1 - 3.41e46T + 1.08e94T^{2} \) |
| 89 | \( 1 - 2.72e47T + 3.31e95T^{2} \) |
| 97 | \( 1 - 7.10e48T + 2.24e97T^{2} \) |
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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
−11.44686811755639176798756119717, −9.825418433583763045905987064751, −8.532856377877836841142659207579, −7.55584319074550835208110200340, −6.45262264956560585830374141843, −5.04775263835487492349885414580, −3.90085191813886207117298311163, −3.15781888198153297835241578775, −2.15044158838848237406064203462, −0.56580599530273756148940753427,
0.56580599530273756148940753427, 2.15044158838848237406064203462, 3.15781888198153297835241578775, 3.90085191813886207117298311163, 5.04775263835487492349885414580, 6.45262264956560585830374141843, 7.55584319074550835208110200340, 8.532856377877836841142659207579, 9.825418433583763045905987064751, 11.44686811755639176798756119717
