| L(s) = 1 | − 3-s + 2·5-s + 9-s + 13-s − 2·15-s + 2·17-s + 4·19-s − 25-s − 27-s + 6·29-s − 2·37-s − 39-s + 6·41-s + 12·43-s + 2·45-s + 4·47-s − 7·49-s − 2·51-s + 6·53-s − 4·57-s + 8·59-s − 2·61-s + 2·65-s − 4·67-s + 12·71-s − 14·73-s + 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.277·13-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.82·43-s + 0.298·45-s + 0.583·47-s − 49-s − 0.280·51-s + 0.824·53-s − 0.529·57-s + 1.04·59-s − 0.256·61-s + 0.248·65-s − 0.488·67-s + 1.42·71-s − 1.63·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.498898867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498898867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−10.52145380756309052093810401297, −9.841123515829091442616864691151, −9.067897300263890057771705983710, −7.905269575500993133277771362341, −6.92038776336112574405112626570, −5.93233764524938909111793115936, −5.35188862470330772210371090221, −4.11794986731058490261504789900, −2.68084777327604974871877197850, −1.20573633415257204749821945410,
1.20573633415257204749821945410, 2.68084777327604974871877197850, 4.11794986731058490261504789900, 5.35188862470330772210371090221, 5.93233764524938909111793115936, 6.92038776336112574405112626570, 7.905269575500993133277771362341, 9.067897300263890057771705983710, 9.841123515829091442616864691151, 10.52145380756309052093810401297
