| L(s) = 1 | − 10.2·3-s − 5·5-s − 25.3·7-s + 77.6·9-s − 38.5·11-s − 60.1·13-s + 51.1·15-s + 62.2·17-s − 11.7·19-s + 258.·21-s + 14.8·23-s + 25·25-s − 517.·27-s − 29·29-s + 37.9·31-s + 393.·33-s + 126.·35-s + 118.·37-s + 615.·39-s + 447.·41-s + 391.·43-s − 388.·45-s + 104.·47-s + 297.·49-s − 636.·51-s − 524.·53-s + 192.·55-s + ⋯ |
| L(s) = 1 | − 1.96·3-s − 0.447·5-s − 1.36·7-s + 2.87·9-s − 1.05·11-s − 1.28·13-s + 0.880·15-s + 0.887·17-s − 0.142·19-s + 2.69·21-s + 0.134·23-s + 0.200·25-s − 3.69·27-s − 0.185·29-s + 0.220·31-s + 2.07·33-s + 0.611·35-s + 0.527·37-s + 2.52·39-s + 1.70·41-s + 1.38·43-s − 1.28·45-s + 0.323·47-s + 0.867·49-s − 1.74·51-s − 1.36·53-s + 0.472·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 10.2T + 27T^{2} \) |
| 7 | \( 1 + 25.3T + 343T^{2} \) |
| 11 | \( 1 + 38.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 14.8T + 1.21e4T^{2} \) |
| 31 | \( 1 - 37.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 118.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 391.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 104.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 524.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 99.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 691.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 462.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 435.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 881.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 43.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 530.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
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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
−9.502892505999533086217128350372, −7.73142142465983467405000346340, −7.23922123534429879540891237230, −6.34083682197021405767930040549, −5.64141522941227902312445598784, −4.90667821512521181259987068105, −3.95754368450771663202925012259, −2.61789890344880446666978331750, −0.78113423144040520703420115014, 0,
0.78113423144040520703420115014, 2.61789890344880446666978331750, 3.95754368450771663202925012259, 4.90667821512521181259987068105, 5.64141522941227902312445598784, 6.34083682197021405767930040549, 7.23922123534429879540891237230, 7.73142142465983467405000346340, 9.502892505999533086217128350372
