L(s) = 1 | − 1.72·2-s − 3-s + 0.973·4-s − 0.0158·5-s + 1.72·6-s + 3.08·7-s + 1.77·8-s + 9-s + 0.0273·10-s − 11-s − 0.973·12-s + 0.101·13-s − 5.32·14-s + 0.0158·15-s − 4.99·16-s − 0.754·17-s − 1.72·18-s − 6.01·19-s − 0.0154·20-s − 3.08·21-s + 1.72·22-s + 7.29·23-s − 1.77·24-s − 4.99·25-s − 0.175·26-s − 27-s + 3.00·28-s + ⋯ |
L(s) = 1 | − 1.21·2-s − 0.577·3-s + 0.486·4-s − 0.00710·5-s + 0.703·6-s + 1.16·7-s + 0.625·8-s + 0.333·9-s + 0.00865·10-s − 0.301·11-s − 0.280·12-s + 0.0281·13-s − 1.42·14-s + 0.00409·15-s − 1.24·16-s − 0.182·17-s − 0.406·18-s − 1.37·19-s − 0.00345·20-s − 0.673·21-s + 0.367·22-s + 1.52·23-s − 0.361·24-s − 0.999·25-s − 0.0343·26-s − 0.192·27-s + 0.568·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 1.72T + 2T^{2} \) |
| 5 | \( 1 + 0.0158T + 5T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 13 | \( 1 - 0.101T + 13T^{2} \) |
| 17 | \( 1 + 0.754T + 17T^{2} \) |
| 19 | \( 1 + 6.01T + 19T^{2} \) |
| 23 | \( 1 - 7.29T + 23T^{2} \) |
| 29 | \( 1 - 2.30T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 + 6.03T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 - 1.22T + 59T^{2} \) |
| 67 | \( 1 - 9.40T + 67T^{2} \) |
| 71 | \( 1 + 9.66T + 71T^{2} \) |
| 73 | \( 1 - 5.87T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 + 6.37T + 97T^{2} \) |
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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
−8.682713254383431353740653386500, −8.179889131619529097146950590057, −7.34151095951755057227216493122, −6.71767751201858531820739852743, −5.48209149745464779920654747319, −4.82462860884360394012457378939, −3.93936225712806281070654525522, −2.21825798597905997329159706244, −1.36511887351668887794095117023, 0,
1.36511887351668887794095117023, 2.21825798597905997329159706244, 3.93936225712806281070654525522, 4.82462860884360394012457378939, 5.48209149745464779920654747319, 6.71767751201858531820739852743, 7.34151095951755057227216493122, 8.179889131619529097146950590057, 8.682713254383431353740653386500