| L(s) = 1 | − 2.69·2-s + 5.25·4-s + 2.61·5-s − 3.45·7-s − 8.77·8-s − 7.03·10-s + 11-s − 0.988·13-s + 9.30·14-s + 13.1·16-s + 5.16·17-s − 1.22·19-s + 13.7·20-s − 2.69·22-s + 8.70·23-s + 1.81·25-s + 2.66·26-s − 18.1·28-s − 8.05·29-s − 6.26·31-s − 17.8·32-s − 13.9·34-s − 9.02·35-s − 2.93·37-s + 3.30·38-s − 22.9·40-s + 0.140·41-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.62·4-s + 1.16·5-s − 1.30·7-s − 3.10·8-s − 2.22·10-s + 0.301·11-s − 0.274·13-s + 2.48·14-s + 3.28·16-s + 1.25·17-s − 0.281·19-s + 3.07·20-s − 0.574·22-s + 1.81·23-s + 0.363·25-s + 0.522·26-s − 3.43·28-s − 1.49·29-s − 1.12·31-s − 3.15·32-s − 2.38·34-s − 1.52·35-s − 0.482·37-s + 0.536·38-s − 3.62·40-s + 0.0219·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 13 | \( 1 + 0.988T + 13T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 23 | \( 1 - 8.70T + 23T^{2} \) |
| 29 | \( 1 + 8.05T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 + 2.93T + 37T^{2} \) |
| 41 | \( 1 - 0.140T + 41T^{2} \) |
| 43 | \( 1 - 0.373T + 43T^{2} \) |
| 47 | \( 1 - 1.80T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 2.83T + 59T^{2} \) |
| 67 | \( 1 + 1.15T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 2.41T + 89T^{2} \) |
| 97 | \( 1 + 1.08T + 97T^{2} \) |
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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
−7.67099951778041188340902180239, −7.19066268141914920220737602035, −6.52164501362449582807328484577, −5.93126904317370282677942552051, −5.29158520891904722425865614072, −3.51060909018206378072919185662, −2.91507173395111422747586651119, −1.97699975190126649167632261262, −1.19961033859544543774686889837, 0,
1.19961033859544543774686889837, 1.97699975190126649167632261262, 2.91507173395111422747586651119, 3.51060909018206378072919185662, 5.29158520891904722425865614072, 5.93126904317370282677942552051, 6.52164501362449582807328484577, 7.19066268141914920220737602035, 7.67099951778041188340902180239
