| L(s) = 1 | + 238.·2-s − 5.90e4·3-s − 2.04e6·4-s − 1.40e7·6-s + 6.18e8·7-s − 9.87e8·8-s + 3.48e9·9-s − 1.54e11·11-s + 1.20e11·12-s − 3.64e11·13-s + 1.47e11·14-s + 4.04e12·16-s + 2.70e12·17-s + 8.32e11·18-s − 5.88e12·19-s − 3.65e13·21-s − 3.68e13·22-s − 3.03e14·23-s + 5.83e13·24-s − 8.69e13·26-s − 2.05e14·27-s − 1.26e15·28-s + 1.45e15·29-s + 1.69e15·31-s + 3.03e15·32-s + 9.12e15·33-s + 6.46e14·34-s + ⋯ |
| L(s) = 1 | + 0.164·2-s − 0.577·3-s − 0.972·4-s − 0.0951·6-s + 0.827·7-s − 0.325·8-s + 0.333·9-s − 1.79·11-s + 0.561·12-s − 0.732·13-s + 0.136·14-s + 0.919·16-s + 0.325·17-s + 0.0549·18-s − 0.220·19-s − 0.477·21-s − 0.296·22-s − 1.53·23-s + 0.187·24-s − 0.120·26-s − 0.192·27-s − 0.804·28-s + 0.640·29-s + 0.370·31-s + 0.476·32-s + 1.03·33-s + 0.0537·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.3948543463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3948543463\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 238.T + 2.09e6T^{2} \) |
| 7 | \( 1 - 6.18e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.54e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 3.64e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.70e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 5.88e12T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.03e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.45e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 1.69e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.82e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 4.21e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.70e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.64e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.18e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 1.67e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 8.99e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.10e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.20e18T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.93e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.30e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.91e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.00e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.27e19T + 5.27e41T^{2} \) |
| 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
−10.48704962194312843629253436934, −9.824815600437478147429292289826, −8.281791199128058303938970460244, −7.70133797957742334470451882812, −5.99737041179507692513669753005, −5.02044436375944103180021400022, −4.51442852049735914155920327208, −2.98710775591235245565334633686, −1.66619580655649589720828231463, −0.26159192660656068599485357724,
0.26159192660656068599485357724, 1.66619580655649589720828231463, 2.98710775591235245565334633686, 4.51442852049735914155920327208, 5.02044436375944103180021400022, 5.99737041179507692513669753005, 7.70133797957742334470451882812, 8.281791199128058303938970460244, 9.824815600437478147429292289826, 10.48704962194312843629253436934
