| L(s) = 1 | + 32·2-s + 1.02e3·4-s − 1.16e4·5-s − 1.68e4·7-s + 3.27e4·8-s − 3.73e5·10-s + 8.09e5·11-s − 3.65e5·13-s − 5.37e5·14-s + 1.04e6·16-s + 6.90e6·17-s + 8.68e5·19-s − 1.19e7·20-s + 2.59e7·22-s + 4.70e6·23-s + 8.73e7·25-s − 1.16e7·26-s − 1.72e7·28-s − 1.62e8·29-s + 1.83e8·31-s + 3.35e7·32-s + 2.21e8·34-s + 1.96e8·35-s − 4.64e8·37-s + 2.78e7·38-s − 3.82e8·40-s − 1.03e9·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.67·5-s − 0.377·7-s + 0.353·8-s − 1.18·10-s + 1.51·11-s − 0.272·13-s − 0.267·14-s + 0.250·16-s + 1.18·17-s + 0.0805·19-s − 0.835·20-s + 1.07·22-s + 0.152·23-s + 1.78·25-s − 0.192·26-s − 0.188·28-s − 1.47·29-s + 1.14·31-s + 0.176·32-s + 0.834·34-s + 0.631·35-s − 1.10·37-s + 0.0569·38-s − 0.590·40-s − 1.39·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 32T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 + 1.16e4T + 4.88e7T^{2} \) |
| 11 | \( 1 - 8.09e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 3.65e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.90e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 8.68e5T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.70e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.62e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.83e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.64e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.03e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.78e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.16e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.50e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + 4.54e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 5.61e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.80e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.69e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.64e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.81e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.65e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.37e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.81e10T + 7.15e21T^{2} \) |
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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
−11.16439217039954677140301975008, −9.754860021322933623180151553029, −8.408455038314789952744390850971, −7.37315160553529663153239388666, −6.48808945173885648107794449683, −4.97204998214140044370803347034, −3.80744711717111778366904741886, −3.29167170206105334298545772942, −1.36388773686654123404986843762, 0,
1.36388773686654123404986843762, 3.29167170206105334298545772942, 3.80744711717111778366904741886, 4.97204998214140044370803347034, 6.48808945173885648107794449683, 7.37315160553529663153239388666, 8.408455038314789952744390850971, 9.754860021322933623180151553029, 11.16439217039954677140301975008
