| L(s) = 1 | + 32·2-s + 243·3-s + 1.02e3·4-s − 5.27e3·5-s + 7.77e3·6-s − 3.02e4·7-s + 3.27e4·8-s + 5.90e4·9-s − 1.68e5·10-s + 1.61e5·11-s + 2.48e5·12-s + 4.36e5·13-s − 9.68e5·14-s − 1.28e6·15-s + 1.04e6·16-s − 5.19e5·17-s + 1.88e6·18-s − 8.72e6·19-s − 5.39e6·20-s − 7.35e6·21-s + 5.15e6·22-s − 3.77e7·23-s + 7.96e6·24-s − 2.10e7·25-s + 1.39e7·26-s + 1.43e7·27-s − 3.09e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.754·5-s + 0.408·6-s − 0.680·7-s + 0.353·8-s + 1/3·9-s − 0.533·10-s + 0.301·11-s + 0.288·12-s + 0.325·13-s − 0.481·14-s − 0.435·15-s + 1/4·16-s − 0.0887·17-s + 0.235·18-s − 0.808·19-s − 0.377·20-s − 0.392·21-s + 0.213·22-s − 1.22·23-s + 0.204·24-s − 0.431·25-s + 0.230·26-s + 0.192·27-s − 0.340·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 - p^{5} T \) |
| 11 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 + 1054 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 4322 p T + p^{11} T^{2} \) |
| 13 | \( 1 - 436342 T + p^{11} T^{2} \) |
| 17 | \( 1 + 30572 p T + p^{11} T^{2} \) |
| 19 | \( 1 + 8726850 T + p^{11} T^{2} \) |
| 23 | \( 1 + 37784208 T + p^{11} T^{2} \) |
| 29 | \( 1 + 204724120 T + p^{11} T^{2} \) |
| 31 | \( 1 - 8717272 T + p^{11} T^{2} \) |
| 37 | \( 1 + 343656494 T + p^{11} T^{2} \) |
| 41 | \( 1 - 364905112 T + p^{11} T^{2} \) |
| 43 | \( 1 + 538304198 T + p^{11} T^{2} \) |
| 47 | \( 1 - 636811256 T + p^{11} T^{2} \) |
| 53 | \( 1 - 1869748502 T + p^{11} T^{2} \) |
| 59 | \( 1 + 7808446140 T + p^{11} T^{2} \) |
| 61 | \( 1 - 1273413682 T + p^{11} T^{2} \) |
| 67 | \( 1 - 3037549956 T + p^{11} T^{2} \) |
| 71 | \( 1 - 21503658952 T + p^{11} T^{2} \) |
| 73 | \( 1 + 9839804298 T + p^{11} T^{2} \) |
| 79 | \( 1 - 10582738690 T + p^{11} T^{2} \) |
| 83 | \( 1 + 67463272528 T + p^{11} T^{2} \) |
| 89 | \( 1 + 69563499550 T + p^{11} T^{2} \) |
| 97 | \( 1 + 15302382914 T + p^{11} T^{2} \) |
| 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
−12.18879165658329244437646051722, −11.05178979712060718968473886924, −9.693597746003168159899996226178, −8.345355454432151940303564349021, −7.18921222660014364896658309927, −5.94638807016687916362989687040, −4.19103324116654269881782210519, −3.43243641503175411484203852841, −1.94054547158428151663959467741, 0,
1.94054547158428151663959467741, 3.43243641503175411484203852841, 4.19103324116654269881782210519, 5.94638807016687916362989687040, 7.18921222660014364896658309927, 8.345355454432151940303564349021, 9.693597746003168159899996226178, 11.05178979712060718968473886924, 12.18879165658329244437646051722
