| L(s) = 1 | − 16·2-s − 224.·3-s + 256·4-s − 29.7·5-s + 3.59e3·6-s − 3.87e3·7-s − 4.09e3·8-s + 3.09e4·9-s + 476.·10-s + 9.46e4·11-s − 5.75e4·12-s + 1.16e5·13-s + 6.20e4·14-s + 6.69e3·15-s + 6.55e4·16-s − 5.50e5·17-s − 4.94e5·18-s − 1.30e5·19-s − 7.62e3·20-s + 8.72e5·21-s − 1.51e6·22-s − 6.07e5·23-s + 9.21e5·24-s − 1.95e6·25-s − 1.86e6·26-s − 2.52e6·27-s − 9.92e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.60·3-s + 0.5·4-s − 0.0213·5-s + 1.13·6-s − 0.610·7-s − 0.353·8-s + 1.57·9-s + 0.0150·10-s + 1.95·11-s − 0.801·12-s + 1.12·13-s + 0.431·14-s + 0.0341·15-s + 0.250·16-s − 1.59·17-s − 1.11·18-s − 0.229·19-s − 0.0106·20-s + 0.978·21-s − 1.37·22-s − 0.452·23-s + 0.566·24-s − 0.999·25-s − 0.798·26-s − 0.915·27-s − 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 3 | \( 1 + 224.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 29.7T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 9.46e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.16e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.50e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 6.07e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.08e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.46e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.22e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.14e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.39e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.77e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.29e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.61e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.02e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.40e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 7.82e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.18e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.06e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.39e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.41e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.72e8T + 7.60e17T^{2} \) |
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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
−13.45959152625848009264755283122, −11.91639737684654766117965124677, −11.36916345442289835760265903972, −10.10855420687804206229247804109, −8.811619999593939704645488599172, −6.63715117599022358110881392637, −6.20437660695332779350420407240, −4.12721804627354224326873669560, −1.35808564030174337460759639887, 0,
1.35808564030174337460759639887, 4.12721804627354224326873669560, 6.20437660695332779350420407240, 6.63715117599022358110881392637, 8.811619999593939704645488599172, 10.10855420687804206229247804109, 11.36916345442289835760265903972, 11.91639737684654766117965124677, 13.45959152625848009264755283122
