| L(s) = 1 | − 2-s + 2.30·3-s + 4-s − 5-s − 2.30·6-s − 2.30·7-s − 8-s + 2.30·9-s + 10-s + 11-s + 2.30·12-s + 2·13-s + 2.30·14-s − 2.30·15-s + 16-s + 1.30·17-s − 2.30·18-s − 5.30·19-s − 20-s − 5.30·21-s − 22-s − 2.30·24-s + 25-s − 2·26-s − 1.60·27-s − 2.30·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.447·5-s − 0.940·6-s − 0.870·7-s − 0.353·8-s + 0.767·9-s + 0.316·10-s + 0.301·11-s + 0.664·12-s + 0.554·13-s + 0.615·14-s − 0.594·15-s + 0.250·16-s + 0.315·17-s − 0.542·18-s − 1.21·19-s − 0.223·20-s − 1.15·21-s − 0.213·22-s − 0.470·24-s + 0.200·25-s − 0.392·26-s − 0.308·27-s − 0.435·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 + 3.90T + 53T^{2} \) |
| 59 | \( 1 + 6.90T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 + 4.30T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 5.39T + 97T^{2} \) |
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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
−8.121266458089428622775280919227, −7.46501680539040955687583292477, −6.60987002000676653552657172740, −6.14278801260259999960049221083, −4.80734769238177234519064914989, −3.72371730019847255543030089874, −3.31307088615920125474725509630, −2.45095554184746616371836932225, −1.47808066300062661625899460050, 0,
1.47808066300062661625899460050, 2.45095554184746616371836932225, 3.31307088615920125474725509630, 3.72371730019847255543030089874, 4.80734769238177234519064914989, 6.14278801260259999960049221083, 6.60987002000676653552657172740, 7.46501680539040955687583292477, 8.121266458089428622775280919227
