| L(s) = 1 | − 4.08e13·2-s − 7.46e22·3-s − 3.79e28·4-s + 2.52e33·5-s + 3.05e36·6-s + 2.11e40·7-s + 3.17e42·8-s + 3.44e45·9-s − 1.03e47·10-s + 2.27e49·11-s + 2.83e51·12-s + 2.29e52·13-s − 8.66e53·14-s − 1.88e56·15-s + 1.37e57·16-s − 1.90e58·17-s − 1.40e59·18-s − 1.95e60·19-s − 9.56e61·20-s − 1.58e63·21-s − 9.32e62·22-s + 6.96e64·23-s − 2.36e65·24-s + 3.83e66·25-s − 9.39e65·26-s − 9.87e67·27-s − 8.03e68·28-s + ⋯ |
| L(s) = 1 | − 0.205·2-s − 1.61·3-s − 0.957·4-s + 1.58·5-s + 0.332·6-s + 1.52·7-s + 0.402·8-s + 1.62·9-s − 0.326·10-s + 0.779·11-s + 1.55·12-s + 0.281·13-s − 0.313·14-s − 2.57·15-s + 0.875·16-s − 0.680·17-s − 0.333·18-s − 0.355·19-s − 1.52·20-s − 2.47·21-s − 0.160·22-s + 1.44·23-s − 0.651·24-s + 1.51·25-s − 0.0577·26-s − 1.01·27-s − 1.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(96-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+95/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(48)\) |
\(\approx\) |
\(1.588737152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.588737152\) |
| \(L(\frac{97}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 4.08e13T + 3.96e28T^{2} \) |
| 3 | \( 1 + 7.46e22T + 2.12e45T^{2} \) |
| 5 | \( 1 - 2.52e33T + 2.52e66T^{2} \) |
| 7 | \( 1 - 2.11e40T + 1.92e80T^{2} \) |
| 11 | \( 1 - 2.27e49T + 8.55e98T^{2} \) |
| 13 | \( 1 - 2.29e52T + 6.67e105T^{2} \) |
| 17 | \( 1 + 1.90e58T + 7.80e116T^{2} \) |
| 19 | \( 1 + 1.95e60T + 3.03e121T^{2} \) |
| 23 | \( 1 - 6.96e64T + 2.31e129T^{2} \) |
| 29 | \( 1 - 3.29e69T + 8.46e138T^{2} \) |
| 31 | \( 1 + 1.05e70T + 4.77e141T^{2} \) |
| 37 | \( 1 - 6.15e73T + 9.53e148T^{2} \) |
| 41 | \( 1 + 5.28e75T + 1.63e153T^{2} \) |
| 43 | \( 1 + 5.20e77T + 1.51e155T^{2} \) |
| 47 | \( 1 + 1.26e79T + 7.06e158T^{2} \) |
| 53 | \( 1 + 3.16e81T + 6.40e163T^{2} \) |
| 59 | \( 1 - 4.60e83T + 1.70e168T^{2} \) |
| 61 | \( 1 - 4.35e84T + 4.03e169T^{2} \) |
| 67 | \( 1 - 1.06e87T + 2.99e173T^{2} \) |
| 71 | \( 1 + 2.68e87T + 7.40e175T^{2} \) |
| 73 | \( 1 + 4.60e88T + 1.03e177T^{2} \) |
| 79 | \( 1 + 9.17e89T + 1.88e180T^{2} \) |
| 83 | \( 1 - 9.48e90T + 2.05e182T^{2} \) |
| 89 | \( 1 + 2.01e92T + 1.55e185T^{2} \) |
| 97 | \( 1 - 4.06e94T + 5.53e188T^{2} \) |
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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
−14.40301181125629488451605724237, −13.08045287362527040732631435054, −11.36922826871394757166148572739, −10.22368079067104240958557910156, −8.778603087441585053811159659945, −6.54154526020928889759816689960, −5.29696037635343740443657691778, −4.63491644229438769425006526118, −1.64723936603496794651549783888, −0.871584909220155590541556923755,
0.871584909220155590541556923755, 1.64723936603496794651549783888, 4.63491644229438769425006526118, 5.29696037635343740443657691778, 6.54154526020928889759816689960, 8.778603087441585053811159659945, 10.22368079067104240958557910156, 11.36922826871394757166148572739, 13.08045287362527040732631435054, 14.40301181125629488451605724237
