| L(s) = 1 | + 4·2-s − 2.51·3-s + 16·4-s + 104.·5-s − 10.0·6-s + 107.·7-s + 64·8-s − 236.·9-s + 418.·10-s − 692.·11-s − 40.2·12-s − 763.·13-s + 431.·14-s − 263.·15-s + 256·16-s + 74.8·17-s − 946.·18-s + 1.13e3·19-s + 1.67e3·20-s − 271.·21-s − 2.77e3·22-s − 160.·24-s + 7.83e3·25-s − 3.05e3·26-s + 1.20e3·27-s + 1.72e3·28-s − 8.08e3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.161·3-s + 0.5·4-s + 1.87·5-s − 0.114·6-s + 0.832·7-s + 0.353·8-s − 0.973·9-s + 1.32·10-s − 1.72·11-s − 0.0806·12-s − 1.25·13-s + 0.588·14-s − 0.302·15-s + 0.250·16-s + 0.0628·17-s − 0.688·18-s + 0.723·19-s + 0.936·20-s − 0.134·21-s − 1.22·22-s − 0.0570·24-s + 2.50·25-s − 0.886·26-s + 0.318·27-s + 0.416·28-s − 1.78·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.51T + 243T^{2} \) |
| 5 | \( 1 - 104.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 107.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 692.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 763.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 74.8T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.13e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 8.08e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 563.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.29e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.72e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.57e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.71e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.18e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.07e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.64e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.55e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.95e4T + 8.58e9T^{2} \) |
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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
−8.797906159682830529792737468759, −7.81059691996544883946720383882, −6.94732245194646183345204999651, −5.78200887162413002571564606743, −5.23313958249785947009111794185, −5.01906020989443615996363847039, −3.10755913816638856128407680362, −2.35606685300579270153562150451, −1.68020225102863261484957530157, 0,
1.68020225102863261484957530157, 2.35606685300579270153562150451, 3.10755913816638856128407680362, 5.01906020989443615996363847039, 5.23313958249785947009111794185, 5.78200887162413002571564606743, 6.94732245194646183345204999651, 7.81059691996544883946720383882, 8.797906159682830529792737468759
