| L(s) = 1 | − 3.14e17·2-s − 1.19e28·3-s − 6.71e34·4-s + 1.37e41·5-s + 3.77e45·6-s − 3.15e49·7-s + 7.34e52·8-s + 7.70e55·9-s − 4.32e58·10-s − 9.61e60·11-s + 8.05e62·12-s + 7.41e64·13-s + 9.91e66·14-s − 1.64e69·15-s − 1.19e70·16-s − 5.06e71·17-s − 2.42e73·18-s − 9.98e74·19-s − 9.22e75·20-s + 3.77e77·21-s + 3.02e78·22-s + 3.00e78·23-s − 8.79e80·24-s + 1.28e82·25-s − 2.33e82·26-s − 1.26e83·27-s + 2.11e84·28-s + ⋯ |
| L(s) = 1 | − 0.771·2-s − 1.46·3-s − 0.404·4-s + 1.77·5-s + 1.13·6-s − 1.14·7-s + 1.08·8-s + 1.15·9-s − 1.36·10-s − 1.15·11-s + 0.593·12-s + 0.506·13-s + 0.886·14-s − 2.60·15-s − 0.432·16-s − 0.528·17-s − 0.894·18-s − 1.55·19-s − 0.715·20-s + 1.68·21-s + 0.889·22-s + 0.0655·23-s − 1.59·24-s + 2.13·25-s − 0.390·26-s − 0.232·27-s + 0.464·28-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 | \( 1 + 3.14e17T + 1.66e35T^{2} \) |
| 3 | \( 1 + 1.19e28T + 6.65e55T^{2} \) |
| 5 | \( 1 - 1.37e41T + 6.01e81T^{2} \) |
| 7 | \( 1 + 3.15e49T + 7.52e98T^{2} \) |
| 11 | \( 1 + 9.61e60T + 6.96e121T^{2} \) |
| 13 | \( 1 - 7.41e64T + 2.14e130T^{2} \) |
| 17 | \( 1 + 5.06e71T + 9.17e143T^{2} \) |
| 19 | \( 1 + 9.98e74T + 4.11e149T^{2} \) |
| 23 | \( 1 - 3.00e78T + 2.09e159T^{2} \) |
| 29 | \( 1 - 2.99e85T + 1.26e171T^{2} \) |
| 31 | \( 1 - 2.53e87T + 3.08e174T^{2} \) |
| 37 | \( 1 + 3.17e91T + 3.01e183T^{2} \) |
| 41 | \( 1 - 1.96e94T + 4.96e188T^{2} \) |
| 43 | \( 1 + 1.68e95T + 1.30e191T^{2} \) |
| 47 | \( 1 - 8.81e97T + 4.31e195T^{2} \) |
| 53 | \( 1 - 2.65e100T + 5.49e201T^{2} \) |
| 59 | \( 1 - 3.52e103T + 1.54e207T^{2} \) |
| 61 | \( 1 - 2.05e103T + 7.64e208T^{2} \) |
| 67 | \( 1 + 9.08e105T + 4.47e213T^{2} \) |
| 71 | \( 1 + 2.51e108T + 3.95e216T^{2} \) |
| 73 | \( 1 + 6.12e108T + 1.02e218T^{2} \) |
| 79 | \( 1 - 2.77e110T + 1.05e222T^{2} \) |
| 83 | \( 1 - 8.51e111T + 3.40e224T^{2} \) |
| 89 | \( 1 - 1.09e114T + 1.19e228T^{2} \) |
| 97 | \( 1 - 1.46e116T + 2.83e232T^{2} \) |
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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
−12.80488264163650042849805142389, −10.52749269428998993791552406149, −10.17081096139238187951905130456, −8.813026899864564586986407117974, −6.61160783653812489160769424985, −5.84994565623930018733017479025, −4.70572538605617275422451532125, −2.38237177115846581823803131084, −0.953036884019109512333656165982, 0,
0.953036884019109512333656165982, 2.38237177115846581823803131084, 4.70572538605617275422451532125, 5.84994565623930018733017479025, 6.61160783653812489160769424985, 8.813026899864564586986407117974, 10.17081096139238187951905130456, 10.52749269428998993791552406149, 12.80488264163650042849805142389
