L(s) = 1 | − 68.1·2-s − 243·3-s + 2.59e3·4-s − 1.23e4·5-s + 1.65e4·6-s + 4.70e4·7-s − 3.75e4·8-s + 5.90e4·9-s + 8.42e5·10-s + 9.95e5·11-s − 6.31e5·12-s − 2.04e6·13-s − 3.20e6·14-s + 3.00e6·15-s − 2.76e6·16-s + 4.93e6·17-s − 4.02e6·18-s − 7.46e6·19-s − 3.21e7·20-s − 1.14e7·21-s − 6.78e7·22-s − 6.09e7·23-s + 9.13e6·24-s + 1.03e8·25-s + 1.39e8·26-s − 1.43e7·27-s + 1.22e8·28-s + ⋯ |
L(s) = 1 | − 1.50·2-s − 0.577·3-s + 1.26·4-s − 1.76·5-s + 0.869·6-s + 1.05·7-s − 0.405·8-s + 0.333·9-s + 2.66·10-s + 1.86·11-s − 0.732·12-s − 1.52·13-s − 1.59·14-s + 1.02·15-s − 0.658·16-s + 0.842·17-s − 0.502·18-s − 0.691·19-s − 2.24·20-s − 0.610·21-s − 2.80·22-s − 1.97·23-s + 0.234·24-s + 2.12·25-s + 2.29·26-s − 0.192·27-s + 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 243T \) |
| 59 | \( 1 - 7.14e8T \) |
good | 2 | \( 1 + 68.1T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.23e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 4.70e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 9.95e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.04e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.93e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 7.46e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 6.09e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.21e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 9.81e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.10e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.11e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 5.23e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.51e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.99e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 5.25e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 4.16e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.59e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 5.20e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.63e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.97e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.53e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 9.77e10T + 7.15e21T^{2} \) |
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
−10.10836091302673282373340987201, −9.097918926169929554316227627796, −7.925550805157353432388336388337, −7.65040909261855035903802177248, −6.52158031886649141715005309644, −4.67269804952743456379715480757, −3.92012143673864871050975369911, −1.91978065685610114640111385908, −0.861470039693743604838652614300, 0,
0.861470039693743604838652614300, 1.91978065685610114640111385908, 3.92012143673864871050975369911, 4.67269804952743456379715480757, 6.52158031886649141715005309644, 7.65040909261855035903802177248, 7.925550805157353432388336388337, 9.097918926169929554316227627796, 10.10836091302673282373340987201