| L(s) = 1 | + 0.219·2-s − 1.95·4-s + 1.55·5-s − 1.60·7-s − 0.868·8-s + 0.340·10-s + 11-s − 4.28·13-s − 0.351·14-s + 3.71·16-s − 1.23·17-s + 3.84·19-s − 3.02·20-s + 0.219·22-s − 7.81·23-s − 2.59·25-s − 0.941·26-s + 3.12·28-s − 7.31·29-s − 4.51·31-s + 2.55·32-s − 0.272·34-s − 2.48·35-s + 1.26·37-s + 0.844·38-s − 1.34·40-s − 4.21·41-s + ⋯ |
| L(s) = 1 | + 0.155·2-s − 0.975·4-s + 0.694·5-s − 0.604·7-s − 0.306·8-s + 0.107·10-s + 0.301·11-s − 1.18·13-s − 0.0939·14-s + 0.928·16-s − 0.300·17-s + 0.882·19-s − 0.677·20-s + 0.0468·22-s − 1.62·23-s − 0.518·25-s − 0.184·26-s + 0.590·28-s − 1.35·29-s − 0.810·31-s + 0.451·32-s − 0.0466·34-s − 0.419·35-s + 0.207·37-s + 0.137·38-s − 0.213·40-s − 0.659·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 0.219T + 2T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 - 3.84T + 19T^{2} \) |
| 23 | \( 1 + 7.81T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 + 4.21T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 5.81T + 47T^{2} \) |
| 53 | \( 1 - 2.75T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.80T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 8.84T + 83T^{2} \) |
| 89 | \( 1 + 9.09T + 89T^{2} \) |
| 97 | \( 1 + 8.22T + 97T^{2} \) |
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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
−9.730443129374046063359597687903, −9.153684810135889410579185786950, −8.053715991227431665278274842469, −7.15463987264097835269544625293, −5.98097438315834265791377738476, −5.38097783376084437584819392194, −4.29743256781591600904054992368, −3.33873972812795889058382473050, −1.95015074235929394309347079557, 0,
1.95015074235929394309347079557, 3.33873972812795889058382473050, 4.29743256781591600904054992368, 5.38097783376084437584819392194, 5.98097438315834265791377738476, 7.15463987264097835269544625293, 8.053715991227431665278274842469, 9.153684810135889410579185786950, 9.730443129374046063359597687903
