| L(s) = 1 | + 4.44e17·2-s + 2.03e27·3-s + 3.11e34·4-s + 7.97e40·5-s + 9.06e44·6-s − 1.30e49·7-s − 5.99e52·8-s − 6.23e55·9-s + 3.54e58·10-s + 8.76e60·11-s + 6.36e61·12-s + 9.17e64·13-s − 5.80e66·14-s + 1.62e68·15-s − 3.18e70·16-s + 3.16e71·17-s − 2.77e73·18-s − 6.40e74·19-s + 2.48e75·20-s − 2.66e76·21-s + 3.89e78·22-s − 5.68e79·23-s − 1.22e80·24-s + 3.35e80·25-s + 4.07e82·26-s − 2.63e83·27-s − 4.07e83·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.250·3-s + 0.187·4-s + 1.02·5-s + 0.272·6-s − 0.475·7-s − 0.885·8-s − 0.937·9-s + 1.11·10-s + 1.05·11-s + 0.0469·12-s + 0.626·13-s − 0.518·14-s + 0.256·15-s − 1.15·16-s + 0.329·17-s − 1.02·18-s − 0.998·19-s + 0.192·20-s − 0.119·21-s + 1.14·22-s − 1.24·23-s − 0.221·24-s + 0.0556·25-s + 0.682·26-s − 0.484·27-s − 0.0893·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(118-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+58.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(59)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{119}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 4.44e17T + 1.66e35T^{2} \) |
| 3 | \( 1 - 2.03e27T + 6.65e55T^{2} \) |
| 5 | \( 1 - 7.97e40T + 6.01e81T^{2} \) |
| 7 | \( 1 + 1.30e49T + 7.52e98T^{2} \) |
| 11 | \( 1 - 8.76e60T + 6.96e121T^{2} \) |
| 13 | \( 1 - 9.17e64T + 2.14e130T^{2} \) |
| 17 | \( 1 - 3.16e71T + 9.17e143T^{2} \) |
| 19 | \( 1 + 6.40e74T + 4.11e149T^{2} \) |
| 23 | \( 1 + 5.68e79T + 2.09e159T^{2} \) |
| 29 | \( 1 + 1.11e85T + 1.26e171T^{2} \) |
| 31 | \( 1 + 1.02e87T + 3.08e174T^{2} \) |
| 37 | \( 1 - 7.11e91T + 3.01e183T^{2} \) |
| 41 | \( 1 - 8.15e93T + 4.96e188T^{2} \) |
| 43 | \( 1 + 5.45e95T + 1.30e191T^{2} \) |
| 47 | \( 1 + 7.07e97T + 4.31e195T^{2} \) |
| 53 | \( 1 - 1.20e101T + 5.49e201T^{2} \) |
| 59 | \( 1 + 6.74e103T + 1.54e207T^{2} \) |
| 61 | \( 1 - 1.25e104T + 7.64e208T^{2} \) |
| 67 | \( 1 + 5.05e106T + 4.47e213T^{2} \) |
| 71 | \( 1 + 1.75e108T + 3.95e216T^{2} \) |
| 73 | \( 1 + 2.55e108T + 1.02e218T^{2} \) |
| 79 | \( 1 + 1.19e111T + 1.05e222T^{2} \) |
| 83 | \( 1 + 1.10e112T + 3.40e224T^{2} \) |
| 89 | \( 1 + 1.93e114T + 1.19e228T^{2} \) |
| 97 | \( 1 + 2.58e116T + 2.83e232T^{2} \) |
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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
−13.01010330876135573768085212984, −11.60035368582069289237931907674, −9.715076526739111614387378088907, −8.632595093734235491287087327427, −6.30394967623822307808402334234, −5.76603590399396771047391351759, −4.14885874010406987951235497989, −3.08029273520848102044919707110, −1.81127454281176306960033876410, 0,
1.81127454281176306960033876410, 3.08029273520848102044919707110, 4.14885874010406987951235497989, 5.76603590399396771047391351759, 6.30394967623822307808402334234, 8.632595093734235491287087327427, 9.715076526739111614387378088907, 11.60035368582069289237931907674, 13.01010330876135573768085212984
