| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 11-s + 12-s − 13-s + 16-s − 17-s − 18-s + 4·19-s − 22-s + 8·23-s − 24-s + 26-s + 27-s + 6·29-s − 8·31-s − 32-s + 33-s + 34-s + 36-s − 6·37-s − 4·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.213·22-s + 1.66·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s − 0.986·37-s − 0.648·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.66734249585505, −12.31427821474817, −11.80287728603538, −11.26932227669965, −10.74552311069723, −10.64219279027489, −9.771640623398514, −9.548416369653226, −9.067580346434998, −8.789668415582357, −8.217911925528092, −7.671898210677554, −7.359453035823442, −6.844490933689324, −6.483419328977523, −5.829005010785489, −5.160574193330137, −4.854950979174042, −4.172453545456119, −3.482624504824764, −3.071162494997567, −2.671776165874107, −1.882944675370047, −1.419498527253178, −0.8345838482532591, 0,
0.8345838482532591, 1.419498527253178, 1.882944675370047, 2.671776165874107, 3.071162494997567, 3.482624504824764, 4.172453545456119, 4.854950979174042, 5.160574193330137, 5.829005010785489, 6.483419328977523, 6.844490933689324, 7.359453035823442, 7.671898210677554, 8.217911925528092, 8.789668415582357, 9.067580346434998, 9.548416369653226, 9.771640623398514, 10.64219279027489, 10.74552311069723, 11.26932227669965, 11.80287728603538, 12.31427821474817, 12.66734249585505
