| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 3.74i·5-s + 2.64·7-s + 2.82i·8-s + 5.29·10-s + 1.41i·11-s − 3.74i·14-s + 4.00·16-s + 3.74i·17-s + 5.29·19-s − 7.48i·20-s + 2.00·22-s − 7.07i·23-s − 9·25-s + ⋯ |
| L(s) = 1 | − 0.999i·2-s − 1.00·4-s + 1.67i·5-s + 0.999·7-s + 1.00i·8-s + 1.67·10-s + 0.426i·11-s − 1.00i·14-s + 1.00·16-s + 0.907i·17-s + 1.21·19-s − 1.67i·20-s + 0.426·22-s − 1.47i·23-s − 1.80·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 3.74iT - 5T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 3.74iT - 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 7.07iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 3.74iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 15.5iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 18.7iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.76259227284873066895634414655, −10.98540404645454842796489917567, −10.49755975149449144682967072218, −9.479772950094533527994623315843, −8.161502421217799917727649709974, −7.24233074393936968529389918812, −5.83225216106266373223867247245, −4.41873339520697452937231567731, −3.18078334605983012449752342670, −1.97703008993395072583230577514,
1.12952791476287571249147936987, 3.96560513896203749463257506335, 5.20088030260710437746340211427, 5.49932079956869419694866465152, 7.37165960470717459832508723330, 8.044363781605078325827204284040, 9.053025572910434592714899788650, 9.554906488960818207839796349220, 11.29289847639873086646251779633, 12.17802256098636169370029093828
