| L(s) = 1 | − 2.19·2-s − 2.47·3-s + 2.81·4-s − 5-s + 5.43·6-s + 3.64·7-s − 1.79·8-s + 3.12·9-s + 2.19·10-s + 11-s − 6.96·12-s − 2.80·13-s − 7.98·14-s + 2.47·15-s − 1.70·16-s + 4.66·17-s − 6.85·18-s − 4.34·19-s − 2.81·20-s − 9.00·21-s − 2.19·22-s − 5.74·23-s + 4.43·24-s + 25-s + 6.15·26-s − 0.301·27-s + 10.2·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 1.42·3-s + 1.40·4-s − 0.447·5-s + 2.21·6-s + 1.37·7-s − 0.633·8-s + 1.04·9-s + 0.694·10-s + 0.301·11-s − 2.01·12-s − 0.778·13-s − 2.13·14-s + 0.638·15-s − 0.425·16-s + 1.13·17-s − 1.61·18-s − 0.995·19-s − 0.629·20-s − 1.96·21-s − 0.467·22-s − 1.19·23-s + 0.904·24-s + 0.200·25-s + 1.20·26-s − 0.0580·27-s + 1.93·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4367162381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4367162381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 3 | \( 1 + 2.47T + 3T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 - 4.66T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 + 5.74T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 + 4.88T + 31T^{2} \) |
| 37 | \( 1 - 9.96T + 37T^{2} \) |
| 41 | \( 1 + 1.64T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 59 | \( 1 - 3.08T + 59T^{2} \) |
| 61 | \( 1 + 4.34T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 79 | \( 1 + 0.195T + 79T^{2} \) |
| 83 | \( 1 + 3.05T + 83T^{2} \) |
| 89 | \( 1 - 7.36T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.357799967779939216601336499043, −7.71286717649940681849510536879, −7.33172260439298621084934511945, −6.33261571161452270785076406254, −5.66186971598399518323618465609, −4.72602829985878585343171966948, −4.17232186198246004919744735111, −2.42068056104813310351141602789, −1.42615495775074085135059684584, −0.56230400169433761299536305830,
0.56230400169433761299536305830, 1.42615495775074085135059684584, 2.42068056104813310351141602789, 4.17232186198246004919744735111, 4.72602829985878585343171966948, 5.66186971598399518323618465609, 6.33261571161452270785076406254, 7.33172260439298621084934511945, 7.71286717649940681849510536879, 8.357799967779939216601336499043
