| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 7-s − 3·8-s + 9-s − 10-s + 11-s − 12-s − 14-s − 15-s − 16-s + 2·17-s + 18-s − 4·19-s + 20-s − 21-s + 22-s − 3·24-s + 25-s + 27-s + 28-s + 6·29-s − 30-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.218·21-s + 0.213·22-s − 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.30238257592488, −12.88641326386380, −12.49755861453971, −12.03341555785287, −11.73739967386692, −10.96521841303680, −10.43730385522223, −10.03837102313933, −9.440069092824382, −9.021131486676533, −8.599430628110787, −8.130846714578539, −7.705943434930058, −6.852526595714781, −6.679330933554179, −5.936959107819487, −5.526077518753959, −4.816544512948817, −4.318395992083873, −4.057677951191269, −3.225486391033027, −3.126413902704580, −2.360754353804629, −1.569028097077995, −0.7645601832375169, 0,
0.7645601832375169, 1.569028097077995, 2.360754353804629, 3.126413902704580, 3.225486391033027, 4.057677951191269, 4.318395992083873, 4.816544512948817, 5.526077518753959, 5.936959107819487, 6.679330933554179, 6.852526595714781, 7.705943434930058, 8.130846714578539, 8.599430628110787, 9.021131486676533, 9.440069092824382, 10.03837102313933, 10.43730385522223, 10.96521841303680, 11.73739967386692, 12.03341555785287, 12.49755861453971, 12.88641326386380, 13.30238257592488
