| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 7-s − 3·8-s + 9-s − 12-s + 2·13-s − 14-s − 16-s − 3·17-s + 18-s − 3·19-s − 21-s + 23-s − 3·24-s + 2·26-s + 27-s + 28-s + 6·29-s + 2·31-s + 5·32-s − 3·34-s − 36-s − 3·37-s − 3·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.688·19-s − 0.218·21-s + 0.208·23-s − 0.612·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.359·31-s + 0.883·32-s − 0.514·34-s − 1/6·36-s − 0.493·37-s − 0.486·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.38116618941892623765348749279, −6.40980575496850752745336635469, −6.16267195545568326960231942385, −5.15353789080674411845571965126, −4.41862671472905323650542316936, −3.99468080407778510635873084509, −3.08142259729905452089295954421, −2.56674634019174147307302396637, −1.31056166996967453559218243902, 0,
1.31056166996967453559218243902, 2.56674634019174147307302396637, 3.08142259729905452089295954421, 3.99468080407778510635873084509, 4.41862671472905323650542316936, 5.15353789080674411845571965126, 6.16267195545568326960231942385, 6.40980575496850752745336635469, 7.38116618941892623765348749279
