| L(s) = 1 | + 14.3·2-s − 243·3-s − 1.84e3·4-s + 4.42e3·5-s − 3.48e3·6-s + 1.68e4·7-s − 5.58e4·8-s + 5.90e4·9-s + 6.35e4·10-s + 3.38e5·11-s + 4.47e5·12-s − 3.71e5·13-s + 2.41e5·14-s − 1.07e6·15-s + 2.96e6·16-s − 3.75e6·17-s + 8.47e5·18-s − 1.16e7·19-s − 8.14e6·20-s − 4.08e6·21-s + 4.85e6·22-s + 5.77e7·23-s + 1.35e7·24-s − 2.92e7·25-s − 5.33e6·26-s − 1.43e7·27-s − 3.09e7·28-s + ⋯ |
| L(s) = 1 | + 0.317·2-s − 0.577·3-s − 0.899·4-s + 0.633·5-s − 0.183·6-s + 0.377·7-s − 0.602·8-s + 0.333·9-s + 0.200·10-s + 0.633·11-s + 0.519·12-s − 0.277·13-s + 0.119·14-s − 0.365·15-s + 0.708·16-s − 0.640·17-s + 0.105·18-s − 1.07·19-s − 0.569·20-s − 0.218·21-s + 0.200·22-s + 1.87·23-s + 0.347·24-s − 0.599·25-s − 0.0880·26-s − 0.192·27-s − 0.339·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 243T \) |
| 7 | \( 1 - 1.68e4T \) |
| 13 | \( 1 + 3.71e5T \) |
| good | 2 | \( 1 - 14.3T + 2.04e3T^{2} \) |
| 5 | \( 1 - 4.42e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 3.38e5T + 2.85e11T^{2} \) |
| 17 | \( 1 + 3.75e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.16e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.77e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.08e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 9.12e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.76e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.17e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.62e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 7.56e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.53e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 1.62e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.26e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.00e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.02e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.14e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.44e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.15e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.29e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 5.73e10T + 7.15e21T^{2} \) |
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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
−9.303796838299969933901104912097, −8.930982603780717479259057304759, −7.49007155062499259580853196076, −6.34697930523043455518885338085, −5.48593266507547010172032809474, −4.64401427136168205973369116588, −3.76075581574016604638485048405, −2.24666920766488032945938277097, −1.07961153584622090385564780461, 0,
1.07961153584622090385564780461, 2.24666920766488032945938277097, 3.76075581574016604638485048405, 4.64401427136168205973369116588, 5.48593266507547010172032809474, 6.34697930523043455518885338085, 7.49007155062499259580853196076, 8.930982603780717479259057304759, 9.303796838299969933901104912097
