| L(s) = 1 | + 1.28e13·2-s − 4.56e20·3-s + 1.02e25·4-s − 3.44e30·5-s − 5.85e33·6-s − 1.04e37·7-s − 1.85e39·8-s − 1.15e41·9-s − 4.42e43·10-s − 5.68e44·11-s − 4.68e45·12-s + 3.85e48·13-s − 1.34e50·14-s + 1.57e51·15-s − 2.54e52·16-s − 2.85e53·17-s − 1.48e54·18-s − 3.24e54·19-s − 3.54e55·20-s + 4.78e57·21-s − 7.30e57·22-s − 1.68e58·23-s + 8.46e59·24-s + 5.40e60·25-s + 4.95e61·26-s + 1.99e62·27-s − 1.07e62·28-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 0.802·3-s + 0.0664·4-s − 1.35·5-s − 0.828·6-s − 1.81·7-s − 0.964·8-s − 0.356·9-s − 1.39·10-s − 0.284·11-s − 0.0532·12-s + 1.34·13-s − 1.87·14-s + 1.08·15-s − 1.06·16-s − 0.852·17-s − 0.368·18-s − 0.0768·19-s − 0.0900·20-s + 1.45·21-s − 0.293·22-s − 0.0979·23-s + 0.773·24-s + 0.836·25-s + 1.39·26-s + 1.08·27-s − 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(88-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+87/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(44)\) |
\(\approx\) |
\(0.06217574835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06217574835\) |
| \(L(\frac{89}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.28e13T + 1.54e26T^{2} \) |
| 3 | \( 1 + 4.56e20T + 3.23e41T^{2} \) |
| 5 | \( 1 + 3.44e30T + 6.46e60T^{2} \) |
| 7 | \( 1 + 1.04e37T + 3.33e73T^{2} \) |
| 11 | \( 1 + 5.68e44T + 3.99e90T^{2} \) |
| 13 | \( 1 - 3.85e48T + 8.18e96T^{2} \) |
| 17 | \( 1 + 2.85e53T + 1.11e107T^{2} \) |
| 19 | \( 1 + 3.24e54T + 1.78e111T^{2} \) |
| 23 | \( 1 + 1.68e58T + 2.95e118T^{2} \) |
| 29 | \( 1 + 3.19e63T + 1.69e127T^{2} \) |
| 31 | \( 1 + 1.41e65T + 5.60e129T^{2} \) |
| 37 | \( 1 + 1.43e68T + 2.71e136T^{2} \) |
| 41 | \( 1 + 1.53e70T + 2.05e140T^{2} \) |
| 43 | \( 1 - 1.73e70T + 1.29e142T^{2} \) |
| 47 | \( 1 + 1.94e72T + 2.96e145T^{2} \) |
| 53 | \( 1 - 6.95e74T + 1.02e150T^{2} \) |
| 59 | \( 1 + 6.04e76T + 1.15e154T^{2} \) |
| 61 | \( 1 + 4.76e77T + 2.10e155T^{2} \) |
| 67 | \( 1 - 1.53e78T + 7.38e158T^{2} \) |
| 71 | \( 1 - 1.51e80T + 1.14e161T^{2} \) |
| 73 | \( 1 + 3.02e80T + 1.28e162T^{2} \) |
| 79 | \( 1 + 4.48e82T + 1.24e165T^{2} \) |
| 83 | \( 1 - 4.99e83T + 9.11e166T^{2} \) |
| 89 | \( 1 + 1.05e85T + 3.95e169T^{2} \) |
| 97 | \( 1 + 4.27e86T + 7.06e172T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.40820618062474407025754969171, −13.34319672817688978764204924976, −12.30354847957580716873104234652, −11.05172728699295186177551798159, −8.872484443319524337695850932173, −6.69801159781273493607135017565, −5.61541317202669718833250014045, −3.95101594194089168369779118674, −3.22021912053991559863468258748, −0.11796714819926213086850200055,
0.11796714819926213086850200055, 3.22021912053991559863468258748, 3.95101594194089168369779118674, 5.61541317202669718833250014045, 6.69801159781273493607135017565, 8.872484443319524337695850932173, 11.05172728699295186177551798159, 12.30354847957580716873104234652, 13.34319672817688978764204924976, 15.40820618062474407025754969171
