| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 111.·5-s − 36·6-s − 226.·7-s − 64·8-s + 81·9-s + 445.·10-s + 144·12-s + 351.·13-s + 904.·14-s − 1.00e3·15-s + 256·16-s + 1.33e3·17-s − 324·18-s − 464.·19-s − 1.78e3·20-s − 2.03e3·21-s − 2.93e3·23-s − 576·24-s + 9.30e3·25-s − 1.40e3·26-s + 729·27-s − 3.61e3·28-s + 523.·29-s + 4.01e3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.99·5-s − 0.408·6-s − 1.74·7-s − 0.353·8-s + 0.333·9-s + 1.41·10-s + 0.288·12-s + 0.576·13-s + 1.23·14-s − 1.15·15-s + 0.250·16-s + 1.12·17-s − 0.235·18-s − 0.294·19-s − 0.997·20-s − 1.00·21-s − 1.15·23-s − 0.204·24-s + 2.97·25-s − 0.407·26-s + 0.192·27-s − 0.871·28-s + 0.115·29-s + 0.814·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 111.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 226.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 351.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.33e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 464.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 523.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 153.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 671.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.69e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 757.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.07e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.95e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.46e4T + 8.58e9T^{2} \) |
| 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
−9.068735130395062758117243981613, −8.337165125790516111356402081162, −7.60102034390966413063323671223, −6.94399941968206695619587634448, −5.93724141937552332918216709267, −4.12202034135996232633767139335, −3.53765545635976872498152599897, −2.76244417710834365567585700605, −0.885289017156360619677877991456, 0,
0.885289017156360619677877991456, 2.76244417710834365567585700605, 3.53765545635976872498152599897, 4.12202034135996232633767139335, 5.93724141937552332918216709267, 6.94399941968206695619587634448, 7.60102034390966413063323671223, 8.337165125790516111356402081162, 9.068735130395062758117243981613
