| L(s) = 1 | − 2·2-s + 4·4-s − 13.0·5-s + 18.7·7-s − 8·8-s + 26.1·10-s − 41.1·11-s + 32.6·13-s − 37.4·14-s + 16·16-s + 76.8·17-s − 164.·19-s − 52.2·20-s + 82.2·22-s − 23·23-s + 45.5·25-s − 65.2·26-s + 74.8·28-s + 42.5·29-s + 246.·31-s − 32·32-s − 153.·34-s − 244.·35-s + 281.·37-s + 329.·38-s + 104.·40-s − 330.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.16·5-s + 1.01·7-s − 0.353·8-s + 0.825·10-s − 1.12·11-s + 0.696·13-s − 0.714·14-s + 0.250·16-s + 1.09·17-s − 1.99·19-s − 0.584·20-s + 0.796·22-s − 0.208·23-s + 0.364·25-s − 0.492·26-s + 0.505·28-s + 0.272·29-s + 1.42·31-s − 0.176·32-s − 0.775·34-s − 1.17·35-s + 1.25·37-s + 1.40·38-s + 0.412·40-s − 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.021473362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.021473362\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 + 13.0T + 125T^{2} \) |
| 7 | \( 1 - 18.7T + 343T^{2} \) |
| 11 | \( 1 + 41.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 164.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 42.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 246.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 281.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 160.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 578.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 621.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 347.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 372.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 884.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 777.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 188.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 200.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 49.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 767.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.62e3T + 9.12e5T^{2} \) |
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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
−10.75684185996385231387495981656, −10.11428511317089534652347133041, −8.490379588627458602911690274716, −8.228520782245684994372514120691, −7.46952114596809753125504749547, −6.20038531055412961488984320298, −4.89241547694343075654710250287, −3.79488714721016924339049415427, −2.31610547360114321025114322917, −0.72210257980414860704509202335,
0.72210257980414860704509202335, 2.31610547360114321025114322917, 3.79488714721016924339049415427, 4.89241547694343075654710250287, 6.20038531055412961488984320298, 7.46952114596809753125504749547, 8.228520782245684994372514120691, 8.490379588627458602911690274716, 10.11428511317089534652347133041, 10.75684185996385231387495981656
