| L(s) = 1 | − 3.51e14·2-s + 6.56e22·3-s + 8.39e28·4-s + 2.79e33·5-s − 2.30e37·6-s + 5.53e39·7-s − 1.55e43·8-s + 2.18e45·9-s − 9.84e47·10-s + 3.60e48·11-s + 5.50e51·12-s + 1.06e53·13-s − 1.94e54·14-s + 1.83e56·15-s + 2.15e57·16-s + 3.05e58·17-s − 7.67e59·18-s − 5.36e58·19-s + 2.35e62·20-s + 3.62e62·21-s − 1.26e63·22-s + 3.27e64·23-s − 1.02e66·24-s + 5.31e66·25-s − 3.75e67·26-s + 4.11e66·27-s + 4.64e68·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 1.42·3-s + 2.11·4-s + 1.76·5-s − 2.51·6-s + 0.398·7-s − 1.97·8-s + 1.02·9-s − 3.11·10-s + 0.123·11-s + 3.01·12-s + 1.30·13-s − 0.704·14-s + 2.51·15-s + 1.37·16-s + 1.09·17-s − 1.81·18-s − 0.00973·19-s + 3.73·20-s + 0.567·21-s − 0.217·22-s + 0.681·23-s − 2.81·24-s + 2.10·25-s − 2.31·26-s + 0.0421·27-s + 0.844·28-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
| \(L(48)\) |
\(\approx\) |
\(2.653823018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.653823018\) |
| \(L(\frac{97}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 3.51e14T + 3.96e28T^{2} \) |
| 3 | \( 1 - 6.56e22T + 2.12e45T^{2} \) |
| 5 | \( 1 - 2.79e33T + 2.52e66T^{2} \) |
| 7 | \( 1 - 5.53e39T + 1.92e80T^{2} \) |
| 11 | \( 1 - 3.60e48T + 8.55e98T^{2} \) |
| 13 | \( 1 - 1.06e53T + 6.67e105T^{2} \) |
| 17 | \( 1 - 3.05e58T + 7.80e116T^{2} \) |
| 19 | \( 1 + 5.36e58T + 3.03e121T^{2} \) |
| 23 | \( 1 - 3.27e64T + 2.31e129T^{2} \) |
| 29 | \( 1 + 3.24e69T + 8.46e138T^{2} \) |
| 31 | \( 1 + 4.63e70T + 4.77e141T^{2} \) |
| 37 | \( 1 - 2.22e74T + 9.53e148T^{2} \) |
| 41 | \( 1 + 8.69e75T + 1.63e153T^{2} \) |
| 43 | \( 1 - 4.86e77T + 1.51e155T^{2} \) |
| 47 | \( 1 + 1.28e79T + 7.06e158T^{2} \) |
| 53 | \( 1 - 4.42e81T + 6.40e163T^{2} \) |
| 59 | \( 1 + 1.94e84T + 1.70e168T^{2} \) |
| 61 | \( 1 + 5.98e84T + 4.03e169T^{2} \) |
| 67 | \( 1 + 6.60e86T + 2.99e173T^{2} \) |
| 71 | \( 1 + 7.94e87T + 7.40e175T^{2} \) |
| 73 | \( 1 + 4.40e88T + 1.03e177T^{2} \) |
| 79 | \( 1 - 1.51e90T + 1.88e180T^{2} \) |
| 83 | \( 1 - 1.48e91T + 2.05e182T^{2} \) |
| 89 | \( 1 - 5.31e92T + 1.55e185T^{2} \) |
| 97 | \( 1 + 2.46e94T + 5.53e188T^{2} \) |
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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
−14.71726616195234943240384292258, −13.42122394589058228771779973484, −10.74109211076878477763231953278, −9.486315639113138263242197090603, −8.888128369253280556929654401549, −7.64011577888796113443112714468, −6.00362971999975526709392878127, −3.03512424519530726223045049450, −1.85034580017060386234762884960, −1.25140968538733896372477535420,
1.25140968538733896372477535420, 1.85034580017060386234762884960, 3.03512424519530726223045049450, 6.00362971999975526709392878127, 7.64011577888796113443112714468, 8.888128369253280556929654401549, 9.486315639113138263242197090603, 10.74109211076878477763231953278, 13.42122394589058228771779973484, 14.71726616195234943240384292258
