| L(s) = 1 | + 6·2-s − 9·3-s + 4·4-s + 6·5-s − 54·6-s − 40·7-s − 168·8-s + 81·9-s + 36·10-s − 564·11-s − 36·12-s − 638·13-s − 240·14-s − 54·15-s − 1.13e3·16-s + 882·17-s + 486·18-s + 24·20-s + 360·21-s − 3.38e3·22-s − 840·23-s + 1.51e3·24-s − 3.08e3·25-s − 3.82e3·26-s − 729·27-s − 160·28-s − 4.63e3·29-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.577·3-s + 1/8·4-s + 0.107·5-s − 0.612·6-s − 0.308·7-s − 0.928·8-s + 1/3·9-s + 0.113·10-s − 1.40·11-s − 0.0721·12-s − 1.04·13-s − 0.327·14-s − 0.0619·15-s − 1.10·16-s + 0.740·17-s + 0.353·18-s + 0.0134·20-s + 0.178·21-s − 1.49·22-s − 0.331·23-s + 0.535·24-s − 0.988·25-s − 1.11·26-s − 0.192·27-s − 0.0385·28-s − 1.02·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8028987812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8028987812\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 3 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 6 T + p^{5} T^{2} \) |
| 7 | \( 1 + 40 T + p^{5} T^{2} \) |
| 11 | \( 1 + 564 T + p^{5} T^{2} \) |
| 13 | \( 1 + 638 T + p^{5} T^{2} \) |
| 17 | \( 1 - 882 T + p^{5} T^{2} \) |
| 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
−9.496836595909973790585042390430, −8.080345409244966479626722466801, −7.38917060485400188180780658399, −6.27266724183700216894881470675, −5.48241881589994775127136969577, −5.05460062120767708136580067217, −4.02609871485109633885335604064, −3.06423705626129388363328242660, −2.07827300670332858444113879625, −0.31498196263593192206757309694,
0.31498196263593192206757309694, 2.07827300670332858444113879625, 3.06423705626129388363328242660, 4.02609871485109633885335604064, 5.05460062120767708136580067217, 5.48241881589994775127136969577, 6.27266724183700216894881470675, 7.38917060485400188180780658399, 8.080345409244966479626722466801, 9.496836595909973790585042390430
