| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s − 10·10-s − 45·11-s + 12·12-s − 77·13-s + 14·14-s − 15·15-s + 16·16-s − 17·17-s + 18·18-s − 27·19-s − 20·20-s + 21·21-s − 90·22-s − 77·23-s + 24·24-s − 100·25-s − 154·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.23·11-s + 0.288·12-s − 1.64·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.326·19-s − 0.223·20-s + 0.218·21-s − 0.872·22-s − 0.698·23-s + 0.204·24-s − 4/5·25-s − 1.16·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| 17 | \( 1 + p T \) |
| good | 5 | \( 1 + p T + p^{3} T^{2} \) |
| 11 | \( 1 + 45 T + p^{3} T^{2} \) |
| 13 | \( 1 + 77 T + p^{3} T^{2} \) |
| 19 | \( 1 + 27 T + p^{3} T^{2} \) |
| 23 | \( 1 + 77 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 18 T + p^{3} T^{2} \) |
| 37 | \( 1 - 196 T + p^{3} T^{2} \) |
| 41 | \( 1 + 7 T + p^{3} T^{2} \) |
| 43 | \( 1 + 317 T + p^{3} T^{2} \) |
| 47 | \( 1 - 338 T + p^{3} T^{2} \) |
| 53 | \( 1 - 78 T + p^{3} T^{2} \) |
| 59 | \( 1 - 586 T + p^{3} T^{2} \) |
| 61 | \( 1 + 236 T + p^{3} T^{2} \) |
| 67 | \( 1 + 244 T + p^{3} T^{2} \) |
| 71 | \( 1 + 772 T + p^{3} T^{2} \) |
| 73 | \( 1 + 482 T + p^{3} T^{2} \) |
| 79 | \( 1 + 366 T + p^{3} T^{2} \) |
| 83 | \( 1 + 654 T + p^{3} T^{2} \) |
| 89 | \( 1 + 4 T + p^{3} T^{2} \) |
| 97 | \( 1 + 60 T + p^{3} T^{2} \) |
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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
−9.782511479528915466270567925142, −8.517601458846229858659938195214, −7.68099279609433710281998933600, −7.21824088937671536285440436825, −5.81641163813665216675103057874, −4.86815548830220441622928607110, −4.09602548793456253298294447272, −2.82554269012915880420613129619, −2.05163920315884260514488462382, 0,
2.05163920315884260514488462382, 2.82554269012915880420613129619, 4.09602548793456253298294447272, 4.86815548830220441622928607110, 5.81641163813665216675103057874, 7.21824088937671536285440436825, 7.68099279609433710281998933600, 8.517601458846229858659938195214, 9.782511479528915466270567925142
