| L(s) = 1 | − 2·2-s + 4·4-s + 4·5-s − 7·7-s − 8·8-s − 8·10-s + 11·11-s + 62·13-s + 14·14-s + 16·16-s + 120·17-s + 118·19-s + 16·20-s − 22·22-s + 188·23-s − 109·25-s − 124·26-s − 28·28-s − 62·29-s − 322·31-s − 32·32-s − 240·34-s − 28·35-s − 198·37-s − 236·38-s − 32·40-s − 48·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.357·5-s − 0.377·7-s − 0.353·8-s − 0.252·10-s + 0.301·11-s + 1.32·13-s + 0.267·14-s + 1/4·16-s + 1.71·17-s + 1.42·19-s + 0.178·20-s − 0.213·22-s + 1.70·23-s − 0.871·25-s − 0.935·26-s − 0.188·28-s − 0.397·29-s − 1.86·31-s − 0.176·32-s − 1.21·34-s − 0.135·35-s − 0.879·37-s − 1.00·38-s − 0.126·40-s − 0.182·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.959764916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.959764916\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 120 T + p^{3} T^{2} \) |
| 19 | \( 1 - 118 T + p^{3} T^{2} \) |
| 23 | \( 1 - 188 T + p^{3} T^{2} \) |
| 29 | \( 1 + 62 T + p^{3} T^{2} \) |
| 31 | \( 1 + 322 T + p^{3} T^{2} \) |
| 37 | \( 1 + 198 T + p^{3} T^{2} \) |
| 41 | \( 1 + 48 T + p^{3} T^{2} \) |
| 43 | \( 1 - 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 326 T + p^{3} T^{2} \) |
| 53 | \( 1 - 482 T + p^{3} T^{2} \) |
| 59 | \( 1 + 400 T + p^{3} T^{2} \) |
| 61 | \( 1 - 70 T + p^{3} T^{2} \) |
| 67 | \( 1 + 124 T + p^{3} T^{2} \) |
| 71 | \( 1 - 712 T + p^{3} T^{2} \) |
| 73 | \( 1 - 304 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1016 T + p^{3} T^{2} \) |
| 83 | \( 1 + 430 T + p^{3} T^{2} \) |
| 89 | \( 1 + 442 T + p^{3} T^{2} \) |
| 97 | \( 1 + 966 T + p^{3} T^{2} \) |
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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
−9.262295826955870393010128758936, −8.573309298306506053601166633165, −7.51478313389125857808517874366, −7.01329131230209464323102718391, −5.78813083030322329809588115604, −5.44526052970390454000993458854, −3.70657700119678983682207829762, −3.12818202546301659080067958149, −1.62604030816331993542260372154, −0.840372668049718799637171130660,
0.840372668049718799637171130660, 1.62604030816331993542260372154, 3.12818202546301659080067958149, 3.70657700119678983682207829762, 5.44526052970390454000993458854, 5.78813083030322329809588115604, 7.01329131230209464323102718391, 7.51478313389125857808517874366, 8.573309298306506053601166633165, 9.262295826955870393010128758936
