L(s) = 1 | − 9.16·2-s − 28.2·3-s + 52.0·4-s + 25·5-s + 259.·6-s + 66.4·7-s − 183.·8-s + 555.·9-s − 229.·10-s − 121·11-s − 1.47e3·12-s + 856.·13-s − 608.·14-s − 706.·15-s + 18.3·16-s − 2.25e3·17-s − 5.09e3·18-s − 361·19-s + 1.30e3·20-s − 1.87e3·21-s + 1.10e3·22-s + 2.09e3·23-s + 5.18e3·24-s + 625·25-s − 7.84e3·26-s − 8.82e3·27-s + 3.45e3·28-s + ⋯ |
L(s) = 1 | − 1.62·2-s − 1.81·3-s + 1.62·4-s + 0.447·5-s + 2.93·6-s + 0.512·7-s − 1.01·8-s + 2.28·9-s − 0.724·10-s − 0.301·11-s − 2.94·12-s + 1.40·13-s − 0.830·14-s − 0.810·15-s + 0.0178·16-s − 1.88·17-s − 3.70·18-s − 0.229·19-s + 0.727·20-s − 0.928·21-s + 0.488·22-s + 0.825·23-s + 1.83·24-s + 0.200·25-s − 2.27·26-s − 2.33·27-s + 0.833·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4298225190\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4298225190\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
good | 2 | \( 1 + 9.16T + 32T^{2} \) |
| 3 | \( 1 + 28.2T + 243T^{2} \) |
| 7 | \( 1 - 66.4T + 1.68e4T^{2} \) |
| 13 | \( 1 - 856.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.25e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.18e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.51e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.92e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.10e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.53e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.35e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.79e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.53e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.79e4T + 8.58e9T^{2} \) |
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
−9.181825003363411713520470954077, −8.589507018734637298902665045808, −7.44233083772489298144000517157, −6.70779350225014415430391899849, −6.08641238381872534438277299913, −5.15596676245557534840833284720, −4.15790763372507656439039853303, −2.12737294106529977215345423472, −1.30439432735679852562697302930, −0.45040129119925312771928707913,
0.45040129119925312771928707913, 1.30439432735679852562697302930, 2.12737294106529977215345423472, 4.15790763372507656439039853303, 5.15596676245557534840833284720, 6.08641238381872534438277299913, 6.70779350225014415430391899849, 7.44233083772489298144000517157, 8.589507018734637298902665045808, 9.181825003363411713520470954077