| L(s) = 1 | + 2.51·2-s + 3-s + 4.32·4-s + 2.51·6-s − 0.514·7-s + 5.83·8-s + 9-s − 0.806·11-s + 4.32·12-s + 13-s − 1.29·14-s + 6.02·16-s − 2.02·17-s + 2.51·18-s + 0.292·19-s − 0.514·21-s − 2.02·22-s + 8.34·23-s + 5.83·24-s + 2.51·26-s + 27-s − 2.22·28-s − 9.64·29-s + 6.22·31-s + 3.48·32-s − 0.806·33-s − 5.09·34-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.577·3-s + 2.16·4-s + 1.02·6-s − 0.194·7-s + 2.06·8-s + 0.333·9-s − 0.243·11-s + 1.24·12-s + 0.277·13-s − 0.345·14-s + 1.50·16-s − 0.491·17-s + 0.592·18-s + 0.0671·19-s − 0.112·21-s − 0.432·22-s + 1.74·23-s + 1.19·24-s + 0.493·26-s + 0.192·27-s − 0.419·28-s − 1.79·29-s + 1.11·31-s + 0.616·32-s − 0.140·33-s − 0.874·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.276412097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.276412097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 7 | \( 1 + 0.514T + 7T^{2} \) |
| 11 | \( 1 + 0.806T + 11T^{2} \) |
| 17 | \( 1 + 2.02T + 17T^{2} \) |
| 19 | \( 1 - 0.292T + 19T^{2} \) |
| 23 | \( 1 - 8.34T + 23T^{2} \) |
| 29 | \( 1 + 9.64T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 + 6.34T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 + 7.96T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 - 4.12T + 59T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 4.73T + 79T^{2} \) |
| 83 | \( 1 + 6.57T + 83T^{2} \) |
| 89 | \( 1 + 0.971T + 89T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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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
−10.21554517063465326828300457992, −9.162088296457688991491867793351, −8.173004249914473129561381472647, −7.07045918280889921766686695792, −6.55093791867771157453852883269, −5.39792894444573144952499763504, −4.73360538062463509669189516803, −3.63980655631311439176910377127, −3.00524490327714908562701507395, −1.84972757953007336854379127830,
1.84972757953007336854379127830, 3.00524490327714908562701507395, 3.63980655631311439176910377127, 4.73360538062463509669189516803, 5.39792894444573144952499763504, 6.55093791867771157453852883269, 7.07045918280889921766686695792, 8.173004249914473129561381472647, 9.162088296457688991491867793351, 10.21554517063465326828300457992
