| L(s) = 1 | + 70.7·2-s + 308.·3-s + 2.95e3·4-s − 4.58e3·5-s + 2.18e4·6-s − 7.40e3·7-s + 6.41e4·8-s − 8.16e4·9-s − 3.24e5·10-s + 9.13e5·12-s − 1.81e5·13-s − 5.23e5·14-s − 1.41e6·15-s − 1.51e6·16-s + 7.15e6·17-s − 5.77e6·18-s − 3.40e6·19-s − 1.35e7·20-s − 2.28e6·21-s − 1.43e7·23-s + 1.98e7·24-s − 2.78e7·25-s − 1.28e7·26-s − 7.99e7·27-s − 2.18e7·28-s − 1.62e8·29-s − 1.00e8·30-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.734·3-s + 1.44·4-s − 0.656·5-s + 1.14·6-s − 0.166·7-s + 0.692·8-s − 0.461·9-s − 1.02·10-s + 1.05·12-s − 0.135·13-s − 0.260·14-s − 0.481·15-s − 0.360·16-s + 1.22·17-s − 0.720·18-s − 0.315·19-s − 0.946·20-s − 0.122·21-s − 0.464·23-s + 0.508·24-s − 0.569·25-s − 0.212·26-s − 1.07·27-s − 0.240·28-s − 1.47·29-s − 0.753·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| good | 2 | \( 1 - 70.7T + 2.04e3T^{2} \) |
| 3 | \( 1 - 308.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 4.58e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 7.40e3T + 1.97e9T^{2} \) |
| 13 | \( 1 + 1.81e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 7.15e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 3.40e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.43e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.62e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.40e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.45e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.29e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 6.24e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 7.79e6T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.62e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 9.14e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 5.14e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.82e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.18e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.57e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.15e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.06e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.77e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 3.99e10T + 7.15e21T^{2} \) |
| 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
−11.33184338799442839254641233772, −9.810495658070141078475045765186, −8.453781968222045888724343833056, −7.45948346614938369557842574736, −6.12807113169330288195916310364, −5.09908573000237068860506565273, −3.78154740166136553924168249306, −3.24386412054440775457243800473, −2.01900411561495859587692772239, 0,
2.01900411561495859587692772239, 3.24386412054440775457243800473, 3.78154740166136553924168249306, 5.09908573000237068860506565273, 6.12807113169330288195916310364, 7.45948346614938369557842574736, 8.453781968222045888724343833056, 9.810495658070141078475045765186, 11.33184338799442839254641233772
