| L(s) = 1 | − 2.58·2-s + 4.70·4-s − 3.46·5-s + 0.848·7-s − 6.99·8-s + 8.96·10-s − 11-s + 3.03·13-s − 2.19·14-s + 8.69·16-s + 4.70·17-s + 2.99·19-s − 16.2·20-s + 2.58·22-s − 7.07·23-s + 7.00·25-s − 7.85·26-s + 3.98·28-s + 4.88·29-s − 5.30·31-s − 8.52·32-s − 12.1·34-s − 2.93·35-s − 7.34·37-s − 7.75·38-s + 24.2·40-s + 7.20·41-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 2.35·4-s − 1.54·5-s + 0.320·7-s − 2.47·8-s + 2.83·10-s − 0.301·11-s + 0.841·13-s − 0.587·14-s + 2.17·16-s + 1.14·17-s + 0.687·19-s − 3.64·20-s + 0.551·22-s − 1.47·23-s + 1.40·25-s − 1.53·26-s + 0.753·28-s + 0.907·29-s − 0.953·31-s − 1.50·32-s − 2.09·34-s − 0.496·35-s − 1.20·37-s − 1.25·38-s + 3.82·40-s + 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 0.848T + 7T^{2} \) |
| 13 | \( 1 - 3.03T + 13T^{2} \) |
| 17 | \( 1 - 4.70T + 17T^{2} \) |
| 19 | \( 1 - 2.99T + 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 + 5.30T + 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 - 7.20T + 41T^{2} \) |
| 43 | \( 1 - 5.10T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 - 4.19T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 1.99T + 61T^{2} \) |
| 67 | \( 1 + 1.18T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 9.62T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
| 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
−7.69336276720973703035979787128, −7.36738969014347427054365400840, −6.38453577692536044294627981269, −5.69302503438061866301742631130, −4.56575893642927245672314699194, −3.60060797385557232080169321370, −3.03390541095412817950349025350, −1.80797682228285116998646778664, −0.955753170582480227335272930215, 0,
0.955753170582480227335272930215, 1.80797682228285116998646778664, 3.03390541095412817950349025350, 3.60060797385557232080169321370, 4.56575893642927245672314699194, 5.69302503438061866301742631130, 6.38453577692536044294627981269, 7.36738969014347427054365400840, 7.69336276720973703035979787128
