| L(s) = 1 | + 60.8·2-s + 1.15e3·3-s − 4.49e3·4-s − 1.89e4·5-s + 7.03e4·6-s − 4.35e5·7-s − 7.71e5·8-s − 2.56e5·9-s − 1.15e6·10-s + 5.94e6·11-s − 5.19e6·12-s − 4.82e6·13-s − 2.65e7·14-s − 2.19e7·15-s − 1.00e7·16-s + 2.22e6·17-s − 1.56e7·18-s − 2.64e8·19-s + 8.53e7·20-s − 5.04e8·21-s + 3.61e8·22-s + 1.65e8·23-s − 8.92e8·24-s − 8.60e8·25-s − 2.93e8·26-s − 2.14e9·27-s + 1.95e9·28-s + ⋯ |
| L(s) = 1 | + 0.671·2-s + 0.915·3-s − 0.548·4-s − 0.543·5-s + 0.615·6-s − 1.40·7-s − 1.04·8-s − 0.161·9-s − 0.364·10-s + 1.01·11-s − 0.502·12-s − 0.277·13-s − 0.940·14-s − 0.497·15-s − 0.150·16-s + 0.0223·17-s − 0.108·18-s − 1.28·19-s + 0.298·20-s − 1.28·21-s + 0.679·22-s + 0.232·23-s − 0.952·24-s − 0.704·25-s − 0.186·26-s − 1.06·27-s + 0.768·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 4.82e6T \) |
| good | 2 | \( 1 - 60.8T + 8.19e3T^{2} \) |
| 3 | \( 1 - 1.15e3T + 1.59e6T^{2} \) |
| 5 | \( 1 + 1.89e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 4.35e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 5.94e6T + 3.45e13T^{2} \) |
| 17 | \( 1 - 2.22e6T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.64e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.65e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 1.58e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 8.27e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.47e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 4.41e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 5.03e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 4.34e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 7.25e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 1.64e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 4.30e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 3.20e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.62e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 2.01e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.05e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.12e10T + 8.87e24T^{2} \) |
| 89 | \( 1 + 6.73e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 4.97e11T + 6.73e25T^{2} \) |
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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
−15.49841216382385037626082883058, −14.39478769488292547066722334645, −13.28555339312779754505184601232, −12.07511829194888021455035535391, −9.639563633153909265598781943839, −8.506386886133861285610417696980, −6.36048682152915547087365042285, −4.10050086220941526401935035933, −2.99751392182267152125492205119, 0,
2.99751392182267152125492205119, 4.10050086220941526401935035933, 6.36048682152915547087365042285, 8.506386886133861285610417696980, 9.639563633153909265598781943839, 12.07511829194888021455035535391, 13.28555339312779754505184601232, 14.39478769488292547066722334645, 15.49841216382385037626082883058
