| L(s) = 1 | + 1.96·2-s + 1.87·4-s + 5-s − 3.02·7-s − 0.240·8-s + 1.96·10-s + 4.30·11-s − 13-s − 5.96·14-s − 4.22·16-s + 0.303·17-s − 6.04·19-s + 1.87·20-s + 8.47·22-s − 2.95·23-s + 25-s − 1.96·26-s − 5.68·28-s − 8.88·29-s + 0.0639·31-s − 7.84·32-s + 0.597·34-s − 3.02·35-s + 11.1·37-s − 11.9·38-s − 0.240·40-s − 8.19·41-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 0.938·4-s + 0.447·5-s − 1.14·7-s − 0.0850·8-s + 0.622·10-s + 1.29·11-s − 0.277·13-s − 1.59·14-s − 1.05·16-s + 0.0735·17-s − 1.38·19-s + 0.419·20-s + 1.80·22-s − 0.615·23-s + 0.200·25-s − 0.386·26-s − 1.07·28-s − 1.64·29-s + 0.0114·31-s − 1.38·32-s + 0.102·34-s − 0.511·35-s + 1.82·37-s − 1.93·38-s − 0.0380·40-s − 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.96T + 2T^{2} \) |
| 7 | \( 1 + 3.02T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 17 | \( 1 - 0.303T + 17T^{2} \) |
| 19 | \( 1 + 6.04T + 19T^{2} \) |
| 23 | \( 1 + 2.95T + 23T^{2} \) |
| 29 | \( 1 + 8.88T + 29T^{2} \) |
| 31 | \( 1 - 0.0639T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 8.19T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 - 6.88T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 9.15T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 2.85T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 7.06T + 83T^{2} \) |
| 89 | \( 1 + 4.26T + 89T^{2} \) |
| 97 | \( 1 - 5.83T + 97T^{2} \) |
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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
−7.54688688057200269462433951481, −6.71302400876664301522990713919, −6.11161158875703805694735616554, −5.92483800167588651202191294314, −4.74656357585266121286038709365, −4.10396370458599270015100053973, −3.50013205053082113496509137713, −2.66215417185522318205825271255, −1.73540558200051382415866947191, 0,
1.73540558200051382415866947191, 2.66215417185522318205825271255, 3.50013205053082113496509137713, 4.10396370458599270015100053973, 4.74656357585266121286038709365, 5.92483800167588651202191294314, 6.11161158875703805694735616554, 6.71302400876664301522990713919, 7.54688688057200269462433951481
