| L(s) = 1 | + 2.25e7·2-s − 5.34e13·4-s + 1.06e17·5-s + 5.68e20·7-s − 1.39e22·8-s + 2.40e24·10-s + 4.66e25·11-s + 1.95e27·13-s + 1.28e28·14-s − 2.83e29·16-s + 1.59e30·17-s − 3.45e31·19-s − 5.69e30·20-s + 1.05e33·22-s + 2.46e33·23-s − 6.42e33·25-s + 4.41e34·26-s − 3.03e34·28-s − 4.27e35·29-s + 2.26e36·31-s + 1.42e36·32-s + 3.60e37·34-s + 6.04e37·35-s − 1.55e38·37-s − 7.78e38·38-s − 1.48e39·40-s + 1.49e39·41-s + ⋯ |
| L(s) = 1 | + 0.951·2-s − 0.0949·4-s + 0.798·5-s + 1.12·7-s − 1.04·8-s + 0.759·10-s + 1.42·11-s + 0.998·13-s + 1.06·14-s − 0.896·16-s + 1.13·17-s − 1.61·19-s − 0.0758·20-s + 1.35·22-s + 1.07·23-s − 0.361·25-s + 0.949·26-s − 0.106·28-s − 0.634·29-s + 0.655·31-s + 0.189·32-s + 1.08·34-s + 0.895·35-s − 0.591·37-s − 1.53·38-s − 0.832·40-s + 0.458·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(25)\) |
\(\approx\) |
\(5.280912111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.280912111\) |
| \(L(\frac{51}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 2.25e7T + 5.62e14T^{2} \) |
| 5 | \( 1 - 1.06e17T + 1.77e34T^{2} \) |
| 7 | \( 1 - 5.68e20T + 2.56e41T^{2} \) |
| 11 | \( 1 - 4.66e25T + 1.06e51T^{2} \) |
| 13 | \( 1 - 1.95e27T + 3.83e54T^{2} \) |
| 17 | \( 1 - 1.59e30T + 1.95e60T^{2} \) |
| 19 | \( 1 + 3.45e31T + 4.55e62T^{2} \) |
| 23 | \( 1 - 2.46e33T + 5.30e66T^{2} \) |
| 29 | \( 1 + 4.27e35T + 4.54e71T^{2} \) |
| 31 | \( 1 - 2.26e36T + 1.19e73T^{2} \) |
| 37 | \( 1 + 1.55e38T + 6.94e76T^{2} \) |
| 41 | \( 1 - 1.49e39T + 1.06e79T^{2} \) |
| 43 | \( 1 - 5.17e39T + 1.09e80T^{2} \) |
| 47 | \( 1 + 5.79e40T + 8.56e81T^{2} \) |
| 53 | \( 1 - 5.51e41T + 3.08e84T^{2} \) |
| 59 | \( 1 - 1.93e42T + 5.91e86T^{2} \) |
| 61 | \( 1 - 2.41e43T + 3.02e87T^{2} \) |
| 67 | \( 1 + 8.25e44T + 3.00e89T^{2} \) |
| 71 | \( 1 - 1.77e45T + 5.14e90T^{2} \) |
| 73 | \( 1 - 6.25e45T + 2.00e91T^{2} \) |
| 79 | \( 1 - 4.72e46T + 9.63e92T^{2} \) |
| 83 | \( 1 + 4.29e46T + 1.08e94T^{2} \) |
| 89 | \( 1 - 7.42e47T + 3.31e95T^{2} \) |
| 97 | \( 1 + 5.90e48T + 2.24e97T^{2} \) |
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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
−12.15571582720678131930468761311, −10.99136660546735732661841256692, −9.373163426395883201682448316261, −8.380489923468563366819181714982, −6.49454269264157930150925693332, −5.60283290045170744769927132159, −4.47025932694401773666537597579, −3.52278068985766152251466400525, −1.95691098176371587331073223603, −0.977525183561231310596045520748,
0.977525183561231310596045520748, 1.95691098176371587331073223603, 3.52278068985766152251466400525, 4.47025932694401773666537597579, 5.60283290045170744769927132159, 6.49454269264157930150925693332, 8.380489923468563366819181714982, 9.373163426395883201682448316261, 10.99136660546735732661841256692, 12.15571582720678131930468761311
