| L(s) = 1 | + 3-s − 5-s − 5.08·7-s + 9-s + 4.17·11-s − 1.08·13-s − 15-s + 0.648·17-s + 2.70·19-s − 5.08·21-s − 7.52·23-s + 25-s + 27-s + 5.90·29-s − 31-s + 4.17·33-s + 5.08·35-s + 0.913·37-s − 1.08·39-s + 2.17·41-s − 8.17·43-s − 45-s + 10.2·47-s + 18.8·49-s + 0.648·51-s + 5.52·53-s − 4.17·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.92·7-s + 0.333·9-s + 1.25·11-s − 0.301·13-s − 0.258·15-s + 0.157·17-s + 0.620·19-s − 1.10·21-s − 1.56·23-s + 0.200·25-s + 0.192·27-s + 1.09·29-s − 0.179·31-s + 0.726·33-s + 0.859·35-s + 0.150·37-s − 0.173·39-s + 0.339·41-s − 1.24·43-s − 0.149·45-s + 1.49·47-s + 2.69·49-s + 0.0907·51-s + 0.758·53-s − 0.562·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 5.08T + 7T^{2} \) |
| 11 | \( 1 - 4.17T + 11T^{2} \) |
| 13 | \( 1 + 1.08T + 13T^{2} \) |
| 17 | \( 1 - 0.648T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 29 | \( 1 - 5.90T + 29T^{2} \) |
| 37 | \( 1 - 0.913T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 8.17T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 + 0.438T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 + 2.96T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
| 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
−7.40778753109173572012781677151, −6.91142475646651734587028612605, −6.26974359568956614732909468392, −5.65831262539133229581415519384, −4.34465530979254364005947437384, −3.86069795207944480913786002575, −3.18867414274944541852562156086, −2.51038764187387803437759237763, −1.21655947750892312251002142585, 0,
1.21655947750892312251002142585, 2.51038764187387803437759237763, 3.18867414274944541852562156086, 3.86069795207944480913786002575, 4.34465530979254364005947437384, 5.65831262539133229581415519384, 6.26974359568956614732909468392, 6.91142475646651734587028612605, 7.40778753109173572012781677151
