| L(s) = 1 | − 85.7·2-s + 234.·3-s + 5.30e3·4-s − 1.50e3·5-s − 2.00e4·6-s − 7.06e3·7-s − 2.78e5·8-s − 1.22e5·9-s + 1.29e5·10-s + 5.98e5·11-s + 1.24e6·12-s − 2.84e5·13-s + 6.06e5·14-s − 3.52e5·15-s + 1.30e7·16-s + 6.02e6·17-s + 1.04e7·18-s − 1.04e7·19-s − 7.98e6·20-s − 1.65e6·21-s − 5.12e7·22-s + 3.07e7·23-s − 6.53e7·24-s − 4.65e7·25-s + 2.43e7·26-s − 7.01e7·27-s − 3.74e7·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.556·3-s + 2.58·4-s − 0.215·5-s − 1.05·6-s − 0.158·7-s − 3.00·8-s − 0.690·9-s + 0.408·10-s + 1.12·11-s + 1.44·12-s − 0.212·13-s + 0.301·14-s − 0.120·15-s + 3.10·16-s + 1.02·17-s + 1.30·18-s − 0.964·19-s − 0.557·20-s − 0.0885·21-s − 2.12·22-s + 0.995·23-s − 1.67·24-s − 0.953·25-s + 0.402·26-s − 0.940·27-s − 0.411·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 - 2.05e7T \) |
| good | 2 | \( 1 + 85.7T + 2.04e3T^{2} \) |
| 3 | \( 1 - 234.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.50e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 7.06e3T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.98e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.84e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.02e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.04e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.07e7T + 9.52e14T^{2} \) |
| 31 | \( 1 - 2.00e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.29e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.33e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.51e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 5.65e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.83e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 8.59e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 1.25e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.31e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.99e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.78e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.61e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.39e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.21e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.18e10T + 7.15e21T^{2} \) |
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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
−14.38927506067912947247367683235, −12.14378141254838401887767827033, −11.04499088988373359604136556513, −9.657427912189066388969121608348, −8.760759543085982668322423480501, −7.71397899481067044100030098968, −6.32595914978853565714014771931, −3.13775074915110377815035613746, −1.57562838929040543115709607227, 0,
1.57562838929040543115709607227, 3.13775074915110377815035613746, 6.32595914978853565714014771931, 7.71397899481067044100030098968, 8.760759543085982668322423480501, 9.657427912189066388969121608348, 11.04499088988373359604136556513, 12.14378141254838401887767827033, 14.38927506067912947247367683235
