| L(s) = 1 | + 4.18·2-s − 14.4·3-s − 14.5·4-s − 25·5-s − 60.4·6-s + 39.8·7-s − 194.·8-s − 34.2·9-s − 104.·10-s + 121·11-s + 209.·12-s + 251.·13-s + 166.·14-s + 361.·15-s − 348.·16-s − 947.·17-s − 143.·18-s + 361·19-s + 362.·20-s − 575.·21-s + 505.·22-s − 3.71e3·23-s + 2.80e3·24-s + 625·25-s + 1.05e3·26-s + 4.00e3·27-s − 578.·28-s + ⋯ |
| L(s) = 1 | + 0.739·2-s − 0.926·3-s − 0.453·4-s − 0.447·5-s − 0.684·6-s + 0.307·7-s − 1.07·8-s − 0.141·9-s − 0.330·10-s + 0.301·11-s + 0.420·12-s + 0.412·13-s + 0.227·14-s + 0.414·15-s − 0.340·16-s − 0.795·17-s − 0.104·18-s + 0.229·19-s + 0.202·20-s − 0.285·21-s + 0.222·22-s − 1.46·23-s + 0.995·24-s + 0.200·25-s + 0.304·26-s + 1.05·27-s − 0.139·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 4.18T + 32T^{2} \) |
| 3 | \( 1 + 14.4T + 243T^{2} \) |
| 7 | \( 1 - 39.8T + 1.68e4T^{2} \) |
| 13 | \( 1 - 251.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 947.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.72e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.83e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.97e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.08e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.12e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.59e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.31e4T + 8.58e9T^{2} \) |
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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
−8.650787457823790122286223474184, −8.073101365249101856343578734112, −6.69207544555470641726918976128, −6.08240679343581817019584424958, −5.25043158227736145601980154265, −4.46717440314353153820693731656, −3.76643373659113313985663359169, −2.55128503841518561007687393156, −0.927769801141481119556434465154, 0,
0.927769801141481119556434465154, 2.55128503841518561007687393156, 3.76643373659113313985663359169, 4.46717440314353153820693731656, 5.25043158227736145601980154265, 6.08240679343581817019584424958, 6.69207544555470641726918976128, 8.073101365249101856343578734112, 8.650787457823790122286223474184
