| L(s) = 1 | − 1.76e17·2-s + 5.15e27·3-s − 1.35e35·4-s − 8.41e40·5-s − 9.08e44·6-s − 3.65e49·7-s + 5.30e52·8-s − 3.99e55·9-s + 1.48e58·10-s − 1.29e60·11-s − 6.96e62·12-s + 2.80e65·13-s + 6.43e66·14-s − 4.34e68·15-s + 1.31e70·16-s + 9.36e71·17-s + 7.03e72·18-s + 5.89e74·19-s + 1.13e76·20-s − 1.88e77·21-s + 2.28e77·22-s + 7.12e79·23-s + 2.73e80·24-s + 1.06e81·25-s − 4.94e82·26-s − 5.49e83·27-s + 4.93e84·28-s + ⋯ |
| L(s) = 1 | − 0.432·2-s + 0.632·3-s − 0.813·4-s − 1.08·5-s − 0.273·6-s − 1.33·7-s + 0.783·8-s − 0.600·9-s + 0.468·10-s − 0.155·11-s − 0.514·12-s + 1.91·13-s + 0.575·14-s − 0.685·15-s + 0.474·16-s + 0.977·17-s + 0.259·18-s + 0.919·19-s + 0.882·20-s − 0.842·21-s + 0.0670·22-s + 1.55·23-s + 0.495·24-s + 0.177·25-s − 0.827·26-s − 1.01·27-s + 1.08·28-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 | \( 1 + 1.76e17T + 1.66e35T^{2} \) |
| 3 | \( 1 - 5.15e27T + 6.65e55T^{2} \) |
| 5 | \( 1 + 8.41e40T + 6.01e81T^{2} \) |
| 7 | \( 1 + 3.65e49T + 7.52e98T^{2} \) |
| 11 | \( 1 + 1.29e60T + 6.96e121T^{2} \) |
| 13 | \( 1 - 2.80e65T + 2.14e130T^{2} \) |
| 17 | \( 1 - 9.36e71T + 9.17e143T^{2} \) |
| 19 | \( 1 - 5.89e74T + 4.11e149T^{2} \) |
| 23 | \( 1 - 7.12e79T + 2.09e159T^{2} \) |
| 29 | \( 1 + 4.88e85T + 1.26e171T^{2} \) |
| 31 | \( 1 + 7.94e86T + 3.08e174T^{2} \) |
| 37 | \( 1 + 4.40e91T + 3.01e183T^{2} \) |
| 41 | \( 1 + 7.00e93T + 4.96e188T^{2} \) |
| 43 | \( 1 - 3.80e95T + 1.30e191T^{2} \) |
| 47 | \( 1 - 3.47e97T + 4.31e195T^{2} \) |
| 53 | \( 1 + 1.98e100T + 5.49e201T^{2} \) |
| 59 | \( 1 + 1.69e103T + 1.54e207T^{2} \) |
| 61 | \( 1 + 1.34e104T + 7.64e208T^{2} \) |
| 67 | \( 1 - 1.09e107T + 4.47e213T^{2} \) |
| 71 | \( 1 - 8.07e107T + 3.95e216T^{2} \) |
| 73 | \( 1 + 3.15e108T + 1.02e218T^{2} \) |
| 79 | \( 1 - 9.49e110T + 1.05e222T^{2} \) |
| 83 | \( 1 + 3.43e112T + 3.40e224T^{2} \) |
| 89 | \( 1 - 2.67e112T + 1.19e228T^{2} \) |
| 97 | \( 1 + 2.02e116T + 2.83e232T^{2} \) |
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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
−12.84133323736640065390746191915, −11.08308823157201430635264302641, −9.440278501851708817444577317898, −8.576856106927813249104130188785, −7.45725530636490408265763000550, −5.63075756492761278001592817269, −3.66653307308572113135144513267, −3.32583011947617435297976952020, −1.03782551078879549704895614971, 0,
1.03782551078879549704895614971, 3.32583011947617435297976952020, 3.66653307308572113135144513267, 5.63075756492761278001592817269, 7.45725530636490408265763000550, 8.576856106927813249104130188785, 9.440278501851708817444577317898, 11.08308823157201430635264302641, 12.84133323736640065390746191915
