| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 9·5-s − 36·6-s − 88·7-s − 64·8-s + 81·9-s − 36·10-s − 315·11-s + 144·12-s − 421·13-s + 352·14-s + 81·15-s + 256·16-s + 289·17-s − 324·18-s + 89·19-s + 144·20-s − 792·21-s + 1.26e3·22-s + 1.38e3·23-s − 576·24-s − 3.04e3·25-s + 1.68e3·26-s + 729·27-s − 1.40e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.160·5-s − 0.408·6-s − 0.678·7-s − 0.353·8-s + 1/3·9-s − 0.113·10-s − 0.784·11-s + 0.288·12-s − 0.690·13-s + 0.479·14-s + 0.0929·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.0565·19-s + 0.0804·20-s − 0.391·21-s + 0.555·22-s + 0.547·23-s − 0.204·24-s − 0.974·25-s + 0.488·26-s + 0.192·27-s − 0.339·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 17 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 9 T + p^{5} T^{2} \) |
| 7 | \( 1 + 88 T + p^{5} T^{2} \) |
| 11 | \( 1 + 315 T + p^{5} T^{2} \) |
| 13 | \( 1 + 421 T + p^{5} T^{2} \) |
| 19 | \( 1 - 89 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1389 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6318 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7870 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11272 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9411 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10945 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1902 T + p^{5} T^{2} \) |
| 53 | \( 1 + 9678 T + p^{5} T^{2} \) |
| 59 | \( 1 - 438 T + p^{5} T^{2} \) |
| 61 | \( 1 - 23396 T + p^{5} T^{2} \) |
| 67 | \( 1 + 10468 T + p^{5} T^{2} \) |
| 71 | \( 1 - 78324 T + p^{5} T^{2} \) |
| 73 | \( 1 - 31286 T + p^{5} T^{2} \) |
| 79 | \( 1 + 29542 T + p^{5} T^{2} \) |
| 83 | \( 1 + 13722 T + p^{5} T^{2} \) |
| 89 | \( 1 - 73620 T + p^{5} T^{2} \) |
| 97 | \( 1 + 55432 T + p^{5} T^{2} \) |
| 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
−12.45809745588163002095012696771, −11.03792296476744710668424775292, −9.916987384775281900573403385142, −9.219492941364415328921623324863, −7.925783513779382631358368792866, −6.99964669203727690175387251981, −5.44755267546559774402219131266, −3.43517514880989920083232184583, −2.05400240149001249737057956983, 0,
2.05400240149001249737057956983, 3.43517514880989920083232184583, 5.44755267546559774402219131266, 6.99964669203727690175387251981, 7.925783513779382631358368792866, 9.219492941364415328921623324863, 9.916987384775281900573403385142, 11.03792296476744710668424775292, 12.45809745588163002095012696771
