| L(s) = 1 | − 5-s − 7-s + 3·11-s − 2·17-s − 19-s + 23-s − 4·25-s − 4·29-s − 31-s + 35-s − 3·37-s + 41-s − 2·43-s + 10·47-s + 49-s − 4·53-s − 3·55-s + 6·59-s − 8·61-s − 3·71-s − 2·73-s − 3·77-s − 10·79-s + 12·83-s + 2·85-s + 89-s + 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.904·11-s − 0.485·17-s − 0.229·19-s + 0.208·23-s − 4/5·25-s − 0.742·29-s − 0.179·31-s + 0.169·35-s − 0.493·37-s + 0.156·41-s − 0.304·43-s + 1.45·47-s + 1/7·49-s − 0.549·53-s − 0.404·55-s + 0.781·59-s − 1.02·61-s − 0.356·71-s − 0.234·73-s − 0.341·77-s − 1.12·79-s + 1.31·83-s + 0.216·85-s + 0.105·89-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.393884155533111351589261287420, −7.54156179391697976598762835379, −6.86646436170972850204094716192, −6.14770834153401816711585370604, −5.30749592692925057804882236557, −4.19848178416693554561881434544, −3.73809241648926817609105211575, −2.62686483289578372262401362707, −1.47072311201103098256458190742, 0,
1.47072311201103098256458190742, 2.62686483289578372262401362707, 3.73809241648926817609105211575, 4.19848178416693554561881434544, 5.30749592692925057804882236557, 6.14770834153401816711585370604, 6.86646436170972850204094716192, 7.54156179391697976598762835379, 8.393884155533111351589261287420
