L(s) = 1 | − 39.7·2-s − 81·3-s + 1.06e3·4-s + 1.15e3·5-s + 3.22e3·6-s − 1.75e3·7-s − 2.21e4·8-s + 6.56e3·9-s − 4.58e4·10-s + 7.41e4·11-s − 8.66e4·12-s + 1.24e5·13-s + 6.97e4·14-s − 9.33e4·15-s + 3.34e5·16-s − 2.82e5·17-s − 2.60e5·18-s + 4.98e5·19-s + 1.23e6·20-s + 1.42e5·21-s − 2.94e6·22-s − 1.32e6·23-s + 1.79e6·24-s − 6.24e5·25-s − 4.95e6·26-s − 5.31e5·27-s − 1.87e6·28-s + ⋯ |
L(s) = 1 | − 1.75·2-s − 0.577·3-s + 2.08·4-s + 0.824·5-s + 1.01·6-s − 0.276·7-s − 1.91·8-s + 0.333·9-s − 1.44·10-s + 1.52·11-s − 1.20·12-s + 1.20·13-s + 0.485·14-s − 0.476·15-s + 1.27·16-s − 0.821·17-s − 0.585·18-s + 0.876·19-s + 1.72·20-s + 0.159·21-s − 2.68·22-s − 0.986·23-s + 1.10·24-s − 0.319·25-s − 2.12·26-s − 0.192·27-s − 0.576·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 + 1.21e7T \) |
good | 2 | \( 1 + 39.7T + 512T^{2} \) |
| 5 | \( 1 - 1.15e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.75e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.41e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.24e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.82e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.98e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.32e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.58e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.69e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.67e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.96e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.80e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 8.57e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.60e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 1.24e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.73e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 9.64e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.37e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.71e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.07e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.70e8T + 7.60e17T^{2} \) |
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
−10.35303695024934011761050835262, −9.301152468801901921891605011107, −8.966506420646242096177555965877, −7.49575840630741266464567660508, −6.49345835491205152629913530514, −5.83192343992460609720665366148, −3.76779436165368033656896667602, −1.91576564241285239245506706839, −1.25418767263186712554002108124, 0,
1.25418767263186712554002108124, 1.91576564241285239245506706839, 3.76779436165368033656896667602, 5.83192343992460609720665366148, 6.49345835491205152629913530514, 7.49575840630741266464567660508, 8.966506420646242096177555965877, 9.301152468801901921891605011107, 10.35303695024934011761050835262