| L(s) = 1 | + 1.77·2-s − 3·3-s − 4.83·4-s − 9.16·5-s − 5.33·6-s + 17.1·7-s − 22.8·8-s + 9·9-s − 16.3·10-s − 11·11-s + 14.4·12-s + 37.9·13-s + 30.5·14-s + 27.4·15-s − 1.98·16-s + 71.5·17-s + 16.0·18-s − 28.2·19-s + 44.2·20-s − 51.4·21-s − 19.5·22-s − 129.·23-s + 68.5·24-s − 40.9·25-s + 67.6·26-s − 27·27-s − 82.8·28-s + ⋯ |
| L(s) = 1 | + 0.629·2-s − 0.577·3-s − 0.604·4-s − 0.819·5-s − 0.363·6-s + 0.925·7-s − 1.00·8-s + 0.333·9-s − 0.515·10-s − 0.301·11-s + 0.348·12-s + 0.810·13-s + 0.582·14-s + 0.473·15-s − 0.0310·16-s + 1.02·17-s + 0.209·18-s − 0.341·19-s + 0.495·20-s − 0.534·21-s − 0.189·22-s − 1.17·23-s + 0.582·24-s − 0.327·25-s + 0.510·26-s − 0.192·27-s − 0.559·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.290755844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.290755844\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 1.77T + 8T^{2} \) |
| 5 | \( 1 + 9.16T + 125T^{2} \) |
| 7 | \( 1 - 17.1T + 343T^{2} \) |
| 13 | \( 1 - 37.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 129.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 74.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 56.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 48.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 231.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 107.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 453.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 417.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 334.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 341.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 65.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 271.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 513.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−8.448657648922492467665588512411, −8.182368456203995436660355719300, −7.26966650214731276629385475284, −6.10364817477105389740727836208, −5.50724189199940078784208650473, −4.70600343962227574199648410195, −4.00442552834201980262142413061, −3.31426498697360889010070412126, −1.73025678092625065644663685566, −0.49907080893468589397222792836,
0.49907080893468589397222792836, 1.73025678092625065644663685566, 3.31426498697360889010070412126, 4.00442552834201980262142413061, 4.70600343962227574199648410195, 5.50724189199940078784208650473, 6.10364817477105389740727836208, 7.26966650214731276629385475284, 8.182368456203995436660355719300, 8.448657648922492467665588512411
