| L(s) = 1 | − 3-s + 4·5-s − 7-s + 9-s + 2·11-s − 4·15-s + 8·17-s − 19-s + 21-s + 6·23-s + 11·25-s − 27-s − 2·29-s + 8·31-s − 2·33-s − 4·35-s − 10·37-s + 2·41-s + 8·43-s + 4·45-s − 8·47-s + 49-s − 8·51-s − 2·53-s + 8·55-s + 57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.03·15-s + 1.94·17-s − 0.229·19-s + 0.218·21-s + 1.25·23-s + 11/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.348·33-s − 0.676·35-s − 1.64·37-s + 0.312·41-s + 1.21·43-s + 0.596·45-s − 1.16·47-s + 1/7·49-s − 1.12·51-s − 0.274·53-s + 1.07·55-s + 0.132·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.764004484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.764004484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−7.999459029265568668646334994554, −7.04303450846839301489876696091, −6.49863421244456238514997794656, −5.87434780378740204065553547875, −5.36608813272941930176848669270, −4.68990489956707774879555198772, −3.47048086745469615917922129631, −2.74176650189019429236281557656, −1.62766242092232036468125467242, −0.983095089851999490187474980896,
0.983095089851999490187474980896, 1.62766242092232036468125467242, 2.74176650189019429236281557656, 3.47048086745469615917922129631, 4.68990489956707774879555198772, 5.36608813272941930176848669270, 5.87434780378740204065553547875, 6.49863421244456238514997794656, 7.04303450846839301489876696091, 7.999459029265568668646334994554
