L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 5·5-s − 6·6-s + 8·7-s + 8·8-s + 9·9-s − 10·10-s − 40·11-s − 12·12-s − 13·13-s + 16·14-s + 15·15-s + 16·16-s + 10·17-s + 18·18-s − 20·20-s − 24·21-s − 80·22-s − 180·23-s − 24·24-s + 25·25-s − 26·26-s − 27·27-s + 32·28-s + 22·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.431·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.09·11-s − 0.288·12-s − 0.277·13-s + 0.305·14-s + 0.258·15-s + 1/4·16-s + 0.142·17-s + 0.235·18-s − 0.223·20-s − 0.249·21-s − 0.775·22-s − 1.63·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.215·28-s + 0.140·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
good | 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 17 | \( 1 - 10 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + 180 T + p^{3} T^{2} \) |
| 29 | \( 1 - 22 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 502 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 + 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 422 T + p^{3} T^{2} \) |
| 59 | \( 1 - 104 T + p^{3} T^{2} \) |
| 61 | \( 1 + 82 T + p^{3} T^{2} \) |
| 67 | \( 1 + 540 T + p^{3} T^{2} \) |
| 71 | \( 1 - 512 T + p^{3} T^{2} \) |
| 73 | \( 1 - 622 T + p^{3} T^{2} \) |
| 79 | \( 1 - 104 T + p^{3} T^{2} \) |
| 83 | \( 1 - 348 T + p^{3} T^{2} \) |
| 89 | \( 1 + 286 T + p^{3} T^{2} \) |
| 97 | \( 1 - 494 T + p^{3} T^{2} \) |
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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
−10.64624912887780791852273429101, −9.845069930263987727403202797143, −8.254943207392752459456258289298, −7.57567457345124432923379560819, −6.45032664061833076437765358742, −5.38134402802442507693668284381, −4.64220874189508973820695461794, −3.41752143594510271961102404213, −1.92697841858852923773612867077, 0,
1.92697841858852923773612867077, 3.41752143594510271961102404213, 4.64220874189508973820695461794, 5.38134402802442507693668284381, 6.45032664061833076437765358742, 7.57567457345124432923379560819, 8.254943207392752459456258289298, 9.845069930263987727403202797143, 10.64624912887780791852273429101