| L(s) = 1 | − 4·2-s + 3·3-s + 8·4-s − 12·6-s + 7·7-s + 9·9-s + 62·11-s + 24·12-s + 62·13-s − 28·14-s − 64·16-s − 84·17-s − 36·18-s + 100·19-s + 21·21-s − 248·22-s + 42·23-s − 248·26-s + 27·27-s + 56·28-s − 10·29-s − 48·31-s + 256·32-s + 186·33-s + 336·34-s + 72·36-s + 246·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s + 1/3·9-s + 1.69·11-s + 0.577·12-s + 1.32·13-s − 0.534·14-s − 16-s − 1.19·17-s − 0.471·18-s + 1.20·19-s + 0.218·21-s − 2.40·22-s + 0.380·23-s − 1.87·26-s + 0.192·27-s + 0.377·28-s − 0.0640·29-s − 0.278·31-s + 1.41·32-s + 0.981·33-s + 1.69·34-s + 1/3·36-s + 1.09·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.478118412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.478118412\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 11 | \( 1 - 62 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 10 T + p^{3} T^{2} \) |
| 31 | \( 1 + 48 T + p^{3} T^{2} \) |
| 37 | \( 1 - 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 248 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 + 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 120 T + p^{3} T^{2} \) |
| 61 | \( 1 - 622 T + p^{3} T^{2} \) |
| 67 | \( 1 + 904 T + p^{3} T^{2} \) |
| 71 | \( 1 + 678 T + p^{3} T^{2} \) |
| 73 | \( 1 - 642 T + p^{3} T^{2} \) |
| 79 | \( 1 - 740 T + p^{3} T^{2} \) |
| 83 | \( 1 + 468 T + p^{3} T^{2} \) |
| 89 | \( 1 - 200 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1266 T + p^{3} T^{2} \) |
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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
−10.18325401999038391039154968845, −9.219963882586487311190234949854, −8.886560282758193777339264128173, −8.068629214436027196043109618282, −7.07434469826342421742639389334, −6.29392028728990508589755482508, −4.57754970795960599493942132167, −3.48287386851205467590984712048, −1.82092043908304859486927278429, −0.990815767946827111038207083206,
0.990815767946827111038207083206, 1.82092043908304859486927278429, 3.48287386851205467590984712048, 4.57754970795960599493942132167, 6.29392028728990508589755482508, 7.07434469826342421742639389334, 8.068629214436027196043109618282, 8.886560282758193777339264128173, 9.219963882586487311190234949854, 10.18325401999038391039154968845
