| L(s) = 1 | + 2-s − 1.61·3-s + 4-s − 5-s − 1.61·6-s − 1.23·7-s + 8-s − 0.381·9-s − 10-s + 11-s − 1.61·12-s + 1.61·13-s − 1.23·14-s + 1.61·15-s + 16-s − 4·17-s − 0.381·18-s − 3.38·19-s − 20-s + 2.00·21-s + 22-s + 2·23-s − 1.61·24-s + 25-s + 1.61·26-s + 5.47·27-s − 1.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.934·3-s + 0.5·4-s − 0.447·5-s − 0.660·6-s − 0.467·7-s + 0.353·8-s − 0.127·9-s − 0.316·10-s + 0.301·11-s − 0.467·12-s + 0.448·13-s − 0.330·14-s + 0.417·15-s + 0.250·16-s − 0.970·17-s − 0.0900·18-s − 0.775·19-s − 0.223·20-s + 0.436·21-s + 0.213·22-s + 0.417·23-s − 0.330·24-s + 0.200·25-s + 0.317·26-s + 1.05·27-s − 0.233·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 8.09T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 - 1.85T + 71T^{2} \) |
| 79 | \( 1 + 7.70T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.09244901223651387246190871433, −6.57085903710413571396356704297, −6.19169046553217325160733925478, −5.40178416038937511395466951056, −4.61363682636263906814555269293, −4.13840947473353777021933825940, −3.17801160425126046733079752689, −2.44542853482354947600392395511, −1.13873520872397807899966750839, 0,
1.13873520872397807899966750839, 2.44542853482354947600392395511, 3.17801160425126046733079752689, 4.13840947473353777021933825940, 4.61363682636263906814555269293, 5.40178416038937511395466951056, 6.19169046553217325160733925478, 6.57085903710413571396356704297, 7.09244901223651387246190871433
