| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 21·5-s − 36·6-s − 143·7-s + 64·8-s + 81·9-s + 84·10-s − 205·11-s − 144·12-s − 78·13-s − 572·14-s − 189·15-s + 256·16-s − 2.12e3·17-s + 324·18-s + 361·19-s + 336·20-s + 1.28e3·21-s − 820·22-s + 20·23-s − 576·24-s − 2.68e3·25-s − 312·26-s − 729·27-s − 2.28e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.375·5-s − 0.408·6-s − 1.10·7-s + 0.353·8-s + 1/3·9-s + 0.265·10-s − 0.510·11-s − 0.288·12-s − 0.128·13-s − 0.779·14-s − 0.216·15-s + 1/4·16-s − 1.78·17-s + 0.235·18-s + 0.229·19-s + 0.187·20-s + 0.636·21-s − 0.361·22-s + 0.00788·23-s − 0.204·24-s − 0.858·25-s − 0.0905·26-s − 0.192·27-s − 0.551·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 19 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 21 T + p^{5} T^{2} \) |
| 7 | \( 1 + 143 T + p^{5} T^{2} \) |
| 11 | \( 1 + 205 T + p^{5} T^{2} \) |
| 13 | \( 1 + 6 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 125 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 20 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1098 T + p^{5} T^{2} \) |
| 37 | \( 1 + 15128 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9400 T + p^{5} T^{2} \) |
| 43 | \( 1 - 20073 T + p^{5} T^{2} \) |
| 47 | \( 1 - 14105 T + p^{5} T^{2} \) |
| 53 | \( 1 - 26386 T + p^{5} T^{2} \) |
| 59 | \( 1 + 224 p T + p^{5} T^{2} \) |
| 61 | \( 1 + 2293 T + p^{5} T^{2} \) |
| 67 | \( 1 - 35976 T + p^{5} T^{2} \) |
| 71 | \( 1 - 10180 T + p^{5} T^{2} \) |
| 73 | \( 1 - 33109 T + p^{5} T^{2} \) |
| 79 | \( 1 + 53888 T + p^{5} T^{2} \) |
| 83 | \( 1 - 75196 T + p^{5} T^{2} \) |
| 89 | \( 1 - 20618 T + p^{5} T^{2} \) |
| 97 | \( 1 + 84130 T + p^{5} T^{2} \) |
| 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
−12.33515644517513844770557007181, −11.16342955837972580519104358677, −10.21855672034439734295079500797, −9.070570997993098241909527825210, −7.25042966658256303915993415785, −6.29965770488420147688724901940, −5.28288355264582617103433309894, −3.83732089274219418487882650939, −2.24426286247591196931915040950, 0,
2.24426286247591196931915040950, 3.83732089274219418487882650939, 5.28288355264582617103433309894, 6.29965770488420147688724901940, 7.25042966658256303915993415785, 9.070570997993098241909527825210, 10.21855672034439734295079500797, 11.16342955837972580519104358677, 12.33515644517513844770557007181
