| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 390·5-s + 216·6-s − 343·7-s + 512·8-s + 729·9-s − 3.12e3·10-s − 388·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s − 1.05e4·15-s + 4.09e3·16-s − 3.72e4·17-s + 5.83e3·18-s + 3.51e4·19-s − 2.49e4·20-s − 9.26e3·21-s − 3.10e3·22-s − 1.02e5·23-s + 1.38e4·24-s + 7.39e4·25-s − 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.39·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.986·10-s − 0.0878·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.805·15-s + 1/4·16-s − 1.84·17-s + 0.235·18-s + 1.17·19-s − 0.697·20-s − 0.218·21-s − 0.0621·22-s − 1.76·23-s + 0.204·24-s + 0.946·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.464885150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.464885150\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 - p^{3} T \) |
| 7 | \( 1 + p^{3} T \) |
| 13 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 + 78 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 388 T + p^{7} T^{2} \) |
| 17 | \( 1 + 37294 T + p^{7} T^{2} \) |
| 19 | \( 1 - 35164 T + p^{7} T^{2} \) |
| 23 | \( 1 + 102980 T + p^{7} T^{2} \) |
| 29 | \( 1 - 224826 T + p^{7} T^{2} \) |
| 31 | \( 1 + 150552 T + p^{7} T^{2} \) |
| 37 | \( 1 - 306058 T + p^{7} T^{2} \) |
| 41 | \( 1 - 784994 T + p^{7} T^{2} \) |
| 43 | \( 1 + 771532 T + p^{7} T^{2} \) |
| 47 | \( 1 - 653976 T + p^{7} T^{2} \) |
| 53 | \( 1 + 6646 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1376600 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1215494 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3041808 T + p^{7} T^{2} \) |
| 71 | \( 1 - 611256 T + p^{7} T^{2} \) |
| 73 | \( 1 - 3531686 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1351792 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4882216 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6893754 T + p^{7} T^{2} \) |
| 97 | \( 1 - 3768126 T + p^{7} T^{2} \) |
| 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
−9.652610957003786662847572155825, −8.597202559325568974251529179192, −7.77478886825009108727656558453, −7.08103789493710362863281259565, −6.07266470092761179161888304233, −4.63299561557641848505644228271, −4.06532565059314959409107099440, −3.13430665660150287474901555369, −2.17615530135503955002689924909, −0.56592606313912103428729292961,
0.56592606313912103428729292961, 2.17615530135503955002689924909, 3.13430665660150287474901555369, 4.06532565059314959409107099440, 4.63299561557641848505644228271, 6.07266470092761179161888304233, 7.08103789493710362863281259565, 7.77478886825009108727656558453, 8.597202559325568974251529179192, 9.652610957003786662847572155825
