| L(s) = 1 | − 14·2-s + 48·3-s + 68·4-s − 125·5-s − 672·6-s + 840·8-s + 117·9-s + 1.75e3·10-s + 172·11-s + 3.26e3·12-s − 3.86e3·13-s − 6.00e3·15-s − 2.04e4·16-s + 1.22e4·17-s − 1.63e3·18-s + 2.59e4·19-s − 8.50e3·20-s − 2.40e3·22-s + 1.29e4·23-s + 4.03e4·24-s + 1.56e4·25-s + 5.40e4·26-s − 9.93e4·27-s − 8.16e4·29-s + 8.40e4·30-s + 1.56e5·31-s + 1.78e5·32-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 1.02·3-s + 0.531·4-s − 0.447·5-s − 1.27·6-s + 0.580·8-s + 0.0534·9-s + 0.553·10-s + 0.0389·11-s + 0.545·12-s − 0.487·13-s − 0.459·15-s − 1.24·16-s + 0.604·17-s − 0.0662·18-s + 0.867·19-s − 0.237·20-s − 0.0482·22-s + 0.222·23-s + 0.595·24-s + 1/5·25-s + 0.603·26-s − 0.971·27-s − 0.621·29-s + 0.568·30-s + 0.945·31-s + 0.965·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p^{3} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 7 p T + p^{7} T^{2} \) |
| 3 | \( 1 - 16 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 172 T + p^{7} T^{2} \) |
| 13 | \( 1 + 3862 T + p^{7} T^{2} \) |
| 17 | \( 1 - 12254 T + p^{7} T^{2} \) |
| 19 | \( 1 - 25940 T + p^{7} T^{2} \) |
| 23 | \( 1 - 564 p T + p^{7} T^{2} \) |
| 29 | \( 1 + 81610 T + p^{7} T^{2} \) |
| 31 | \( 1 - 156888 T + p^{7} T^{2} \) |
| 37 | \( 1 - 110126 T + p^{7} T^{2} \) |
| 41 | \( 1 + 467882 T + p^{7} T^{2} \) |
| 43 | \( 1 + 499208 T + p^{7} T^{2} \) |
| 47 | \( 1 - 396884 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1280498 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1337420 T + p^{7} T^{2} \) |
| 61 | \( 1 - 923978 T + p^{7} T^{2} \) |
| 67 | \( 1 + 797304 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5103392 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4267478 T + p^{7} T^{2} \) |
| 79 | \( 1 + 960 T + p^{7} T^{2} \) |
| 83 | \( 1 + 6140832 T + p^{7} T^{2} \) |
| 89 | \( 1 + 2010570 T + p^{7} T^{2} \) |
| 97 | \( 1 - 4881934 T + p^{7} T^{2} \) |
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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
−9.909056029293540990281971831290, −9.374220021716801770704449003461, −8.339161688332284537693126638853, −7.87142702508686580996085408640, −6.91093694032112156314716015523, −5.14451159723646969101501231111, −3.72830607343629680673902191389, −2.57785694791060638973223193227, −1.27401742909994405236416983669, 0,
1.27401742909994405236416983669, 2.57785694791060638973223193227, 3.72830607343629680673902191389, 5.14451159723646969101501231111, 6.91093694032112156314716015523, 7.87142702508686580996085408640, 8.339161688332284537693126638853, 9.374220021716801770704449003461, 9.909056029293540990281971831290
