| L(s) = 1 | − 6·2-s + 4·4-s − 6·5-s + 40·7-s + 168·8-s + 36·10-s − 638·13-s − 240·14-s − 1.13e3·16-s + 882·17-s + 556·19-s − 24·20-s + 840·23-s − 3.08e3·25-s + 3.82e3·26-s + 160·28-s + 4.63e3·29-s + 4.40e3·31-s + 1.44e3·32-s − 5.29e3·34-s − 240·35-s − 2.41e3·37-s − 3.33e3·38-s − 1.00e3·40-s − 6.87e3·41-s − 9.64e3·43-s − 5.04e3·46-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s − 0.107·5-s + 0.308·7-s + 0.928·8-s + 0.113·10-s − 1.04·13-s − 0.327·14-s − 1.10·16-s + 0.740·17-s + 0.353·19-s − 0.0134·20-s + 0.331·23-s − 0.988·25-s + 1.11·26-s + 0.0385·28-s + 1.02·29-s + 0.822·31-s + 0.248·32-s − 0.785·34-s − 0.0331·35-s − 0.289·37-s − 0.374·38-s − 0.0996·40-s − 0.638·41-s − 0.795·43-s − 0.351·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 3 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 6 T + p^{5} T^{2} \) |
| 7 | \( 1 - 40 T + p^{5} T^{2} \) |
| 13 | \( 1 + 638 T + p^{5} T^{2} \) |
| 17 | \( 1 - 882 T + p^{5} T^{2} \) |
| 19 | \( 1 - 556 T + p^{5} T^{2} \) |
| 23 | \( 1 - 840 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4638 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4400 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2410 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6870 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9644 T + p^{5} T^{2} \) |
| 47 | \( 1 - 18672 T + p^{5} T^{2} \) |
| 53 | \( 1 + 33750 T + p^{5} T^{2} \) |
| 59 | \( 1 - 18084 T + p^{5} T^{2} \) |
| 61 | \( 1 + 39758 T + p^{5} T^{2} \) |
| 67 | \( 1 + 23068 T + p^{5} T^{2} \) |
| 71 | \( 1 - 4248 T + p^{5} T^{2} \) |
| 73 | \( 1 - 41110 T + p^{5} T^{2} \) |
| 79 | \( 1 + 21920 T + p^{5} T^{2} \) |
| 83 | \( 1 - 82452 T + p^{5} T^{2} \) |
| 89 | \( 1 - 94086 T + p^{5} T^{2} \) |
| 97 | \( 1 - 49442 T + p^{5} T^{2} \) |
| 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
−8.725669886815912494710075927487, −7.930346251862378454689916951129, −7.45879507317093001363523411247, −6.43096361441368098056501479854, −5.15707954371619845050781550158, −4.52381731368620141040146953534, −3.23517266922768788281505000262, −2.00400791790261610813854684311, −1.00113098118788451784834455038, 0,
1.00113098118788451784834455038, 2.00400791790261610813854684311, 3.23517266922768788281505000262, 4.52381731368620141040146953534, 5.15707954371619845050781550158, 6.43096361441368098056501479854, 7.45879507317093001363523411247, 7.930346251862378454689916951129, 8.725669886815912494710075927487
