| L(s) = 1 | − 16·2-s + 62.2·3-s + 256·4-s − 420.·5-s − 995.·6-s + 1.75e3·7-s − 4.09e3·8-s − 1.58e4·9-s + 6.72e3·10-s + 2.53e4·11-s + 1.59e4·12-s − 6.99e3·13-s − 2.80e4·14-s − 2.61e4·15-s + 6.55e4·16-s + 2.40e5·17-s + 2.52e5·18-s − 1.30e5·19-s − 1.07e5·20-s + 1.09e5·21-s − 4.05e5·22-s − 2.31e6·23-s − 2.54e5·24-s − 1.77e6·25-s + 1.11e5·26-s − 2.20e6·27-s + 4.48e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.443·3-s + 0.5·4-s − 0.300·5-s − 0.313·6-s + 0.275·7-s − 0.353·8-s − 0.803·9-s + 0.212·10-s + 0.521·11-s + 0.221·12-s − 0.0678·13-s − 0.194·14-s − 0.133·15-s + 0.250·16-s + 0.698·17-s + 0.567·18-s − 0.229·19-s − 0.150·20-s + 0.122·21-s − 0.368·22-s − 1.72·23-s − 0.156·24-s − 0.909·25-s + 0.0480·26-s − 0.800·27-s + 0.137·28-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 | \( 1 - 62.2T + 1.96e4T^{2} \) |
| 5 | \( 1 + 420.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.75e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.53e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.99e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.40e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 2.31e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.57e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.59e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.09e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.36e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 5.70e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.95e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.81e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.83e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 9.90e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.26e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.71e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.21e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 9.20e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.27e9T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.98139098458629126486070428067, −12.20966176231991608067352030159, −11.20759619341109734133043041513, −9.758615537894340220968745571513, −8.558661561098450850884668100867, −7.56615790628752382989532129469, −5.83641717365814929891329289892, −3.65208497007181374866040355129, −1.92433902697934951331337668580, 0,
1.92433902697934951331337668580, 3.65208497007181374866040355129, 5.83641717365814929891329289892, 7.56615790628752382989532129469, 8.558661561098450850884668100867, 9.758615537894340220968745571513, 11.20759619341109734133043041513, 12.20966176231991608067352030159, 13.98139098458629126486070428067
