| L(s) = 1 | − 1.27e17·2-s + 8.60e27·3-s − 1.49e35·4-s + 5.94e40·5-s − 1.09e45·6-s + 4.46e49·7-s + 4.03e52·8-s + 7.42e54·9-s − 7.58e57·10-s − 4.86e60·11-s − 1.28e63·12-s − 1.65e65·13-s − 5.69e66·14-s + 5.11e68·15-s + 1.97e70·16-s − 1.52e72·17-s − 9.46e71·18-s − 2.16e74·19-s − 8.91e75·20-s + 3.84e77·21-s + 6.20e77·22-s + 3.41e79·23-s + 3.46e80·24-s − 2.48e81·25-s + 2.10e82·26-s − 5.08e83·27-s − 6.69e84·28-s + ⋯ |
| L(s) = 1 | − 0.313·2-s + 1.05·3-s − 0.902·4-s + 0.766·5-s − 0.330·6-s + 1.62·7-s + 0.595·8-s + 0.111·9-s − 0.239·10-s − 0.583·11-s − 0.950·12-s − 1.12·13-s − 0.509·14-s + 0.808·15-s + 0.715·16-s − 1.59·17-s − 0.0349·18-s − 0.337·19-s − 0.691·20-s + 1.71·21-s + 0.182·22-s + 0.744·23-s + 0.627·24-s − 0.412·25-s + 0.353·26-s − 0.936·27-s − 1.46·28-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 | \( 1 + 1.27e17T + 1.66e35T^{2} \) |
| 3 | \( 1 - 8.60e27T + 6.65e55T^{2} \) |
| 5 | \( 1 - 5.94e40T + 6.01e81T^{2} \) |
| 7 | \( 1 - 4.46e49T + 7.52e98T^{2} \) |
| 11 | \( 1 + 4.86e60T + 6.96e121T^{2} \) |
| 13 | \( 1 + 1.65e65T + 2.14e130T^{2} \) |
| 17 | \( 1 + 1.52e72T + 9.17e143T^{2} \) |
| 19 | \( 1 + 2.16e74T + 4.11e149T^{2} \) |
| 23 | \( 1 - 3.41e79T + 2.09e159T^{2} \) |
| 29 | \( 1 - 2.34e85T + 1.26e171T^{2} \) |
| 31 | \( 1 + 2.97e87T + 3.08e174T^{2} \) |
| 37 | \( 1 - 2.20e91T + 3.01e183T^{2} \) |
| 41 | \( 1 + 1.03e93T + 4.96e188T^{2} \) |
| 43 | \( 1 - 5.21e95T + 1.30e191T^{2} \) |
| 47 | \( 1 + 6.36e97T + 4.31e195T^{2} \) |
| 53 | \( 1 + 9.00e100T + 5.49e201T^{2} \) |
| 59 | \( 1 - 2.04e103T + 1.54e207T^{2} \) |
| 61 | \( 1 - 1.30e104T + 7.64e208T^{2} \) |
| 67 | \( 1 + 5.10e106T + 4.47e213T^{2} \) |
| 71 | \( 1 + 2.64e108T + 3.95e216T^{2} \) |
| 73 | \( 1 - 8.88e108T + 1.02e218T^{2} \) |
| 79 | \( 1 + 5.52e109T + 1.05e222T^{2} \) |
| 83 | \( 1 - 1.11e112T + 3.40e224T^{2} \) |
| 89 | \( 1 + 1.64e114T + 1.19e228T^{2} \) |
| 97 | \( 1 + 2.61e116T + 2.83e232T^{2} \) |
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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
−13.07911961624208947043638503854, −10.94156022667828374390598550604, −9.444609049515265661689209059383, −8.558625023312717814965919101194, −7.58508616294608422276694460190, −5.29172944221509281869600141291, −4.35264451308832202408857342812, −2.49800976588505039494979511779, −1.67245587722051495666385498863, 0,
1.67245587722051495666385498863, 2.49800976588505039494979511779, 4.35264451308832202408857342812, 5.29172944221509281869600141291, 7.58508616294608422276694460190, 8.558625023312717814965919101194, 9.444609049515265661689209059383, 10.94156022667828374390598550604, 13.07911961624208947043638503854
