| L(s) = 1 | − 1.35·2-s − 0.175·4-s − 1.59·7-s + 2.93·8-s − 3.33·11-s − 7.05·13-s + 2.15·14-s − 3.61·16-s + 4.09·17-s + 0.567·19-s + 4.50·22-s + 6.30·23-s + 9.52·26-s + 0.279·28-s + 2.78·29-s − 0.995·31-s − 0.988·32-s − 5.53·34-s + 3.55·37-s − 0.766·38-s − 1.16·41-s − 0.117·43-s + 0.584·44-s − 8.51·46-s + 7.64·47-s − 4.45·49-s + 1.23·52-s + ⋯ |
| L(s) = 1 | − 0.955·2-s − 0.0876·4-s − 0.603·7-s + 1.03·8-s − 1.00·11-s − 1.95·13-s + 0.576·14-s − 0.904·16-s + 0.993·17-s + 0.130·19-s + 0.959·22-s + 1.31·23-s + 1.86·26-s + 0.0528·28-s + 0.516·29-s − 0.178·31-s − 0.174·32-s − 0.948·34-s + 0.584·37-s − 0.124·38-s − 0.181·41-s − 0.0178·43-s + 0.0880·44-s − 1.25·46-s + 1.11·47-s − 0.636·49-s + 0.171·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 + 7.05T + 13T^{2} \) |
| 17 | \( 1 - 4.09T + 17T^{2} \) |
| 19 | \( 1 - 0.567T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 + 0.995T + 31T^{2} \) |
| 37 | \( 1 - 3.55T + 37T^{2} \) |
| 41 | \( 1 + 1.16T + 41T^{2} \) |
| 43 | \( 1 + 0.117T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 - 0.523T + 53T^{2} \) |
| 59 | \( 1 + 0.983T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 5.55T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 5.02T + 83T^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 - 1.70T + 97T^{2} \) |
| 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
−7.82327278659781543397200404474, −7.28379594718014289510976195830, −6.70602428458267031342086296325, −5.32927079281889558265067583063, −5.12887549050490496600634733201, −4.12501088483425810225363986744, −2.98386264132959046967222499832, −2.34769278117501035282396288534, −0.983317184296303353983659568152, 0,
0.983317184296303353983659568152, 2.34769278117501035282396288534, 2.98386264132959046967222499832, 4.12501088483425810225363986744, 5.12887549050490496600634733201, 5.32927079281889558265067583063, 6.70602428458267031342086296325, 7.28379594718014289510976195830, 7.82327278659781543397200404474
