| L(s) = 1 | + 23.4·2-s − 1.50e3·4-s + 3.21e3·5-s + 371.·7-s − 8.30e4·8-s + 7.51e4·10-s + 3.81e5·11-s + 3.12e5·13-s + 8.70e3·14-s + 1.12e6·16-s + 3.99e6·17-s − 1.78e7·19-s − 4.81e6·20-s + 8.92e6·22-s + 4.35e7·23-s − 3.85e7·25-s + 7.30e6·26-s − 5.57e5·28-s − 1.19e8·29-s + 2.08e8·31-s + 1.96e8·32-s + 9.35e7·34-s + 1.19e6·35-s − 7.64e8·37-s − 4.18e8·38-s − 2.66e8·40-s − 6.66e8·41-s + ⋯ |
| L(s) = 1 | + 0.517·2-s − 0.732·4-s + 0.459·5-s + 0.00836·7-s − 0.896·8-s + 0.237·10-s + 0.713·11-s + 0.233·13-s + 0.00432·14-s + 0.268·16-s + 0.682·17-s − 1.65·19-s − 0.336·20-s + 0.369·22-s + 1.41·23-s − 0.788·25-s + 0.120·26-s − 0.00612·28-s − 1.07·29-s + 1.30·31-s + 1.03·32-s + 0.353·34-s + 0.00384·35-s − 1.81·37-s − 0.856·38-s − 0.411·40-s − 0.899·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 23.4T + 2.04e3T^{2} \) |
| 5 | \( 1 - 3.21e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 371.T + 1.97e9T^{2} \) |
| 11 | \( 1 - 3.81e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 3.12e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.99e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.78e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.35e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.19e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.08e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 7.64e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 6.66e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.79e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.02e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.41e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 4.02e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 7.72e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 2.85e7T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.72e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 5.25e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.87e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.77e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.62e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 3.52e10T + 7.15e21T^{2} \) |
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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
−11.78986427951576016488653685226, −10.38079746446772104817822525901, −9.275340740351081177449486253626, −8.382306707095298340596618270278, −6.66187847913896212747932006373, −5.57041164952565022190848069742, −4.39586185582340307741536145341, −3.24393569257410695369118942152, −1.54549616300563311069949253405, 0,
1.54549616300563311069949253405, 3.24393569257410695369118942152, 4.39586185582340307741536145341, 5.57041164952565022190848069742, 6.66187847913896212747932006373, 8.382306707095298340596618270278, 9.275340740351081177449486253626, 10.38079746446772104817822525901, 11.78986427951576016488653685226
