| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 2·11-s − 2·13-s + 16-s − 6·17-s + 3·18-s + 2·19-s − 2·22-s + 23-s − 5·25-s + 2·26-s + 2·29-s − 8·31-s − 32-s + 6·34-s − 3·36-s + 4·37-s − 2·38-s + 6·41-s + 2·43-s + 2·44-s − 46-s − 7·49-s + 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 0.603·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.458·19-s − 0.426·22-s + 0.208·23-s − 25-s + 0.392·26-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 1/2·36-s + 0.657·37-s − 0.324·38-s + 0.937·41-s + 0.304·43-s + 0.301·44-s − 0.147·46-s − 49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8733587421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8733587421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.136255743189544970253673972909, −7.45874998251234741861678003789, −6.73820977248035087534882054690, −6.07413394149159065098784541999, −5.35929050554392728258872019661, −4.41659294228689110102232518535, −3.52251830803368942520302850264, −2.57633528057070295510804316455, −1.87419581676770393357237438018, −0.52491069196242228043430119310,
0.52491069196242228043430119310, 1.87419581676770393357237438018, 2.57633528057070295510804316455, 3.52251830803368942520302850264, 4.41659294228689110102232518535, 5.35929050554392728258872019661, 6.07413394149159065098784541999, 6.73820977248035087534882054690, 7.45874998251234741861678003789, 8.136255743189544970253673972909
