| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s + 9-s − 5·11-s + 12-s + 2·13-s + 3·14-s + 16-s − 17-s − 18-s + 19-s − 3·21-s + 5·22-s − 6·23-s − 24-s − 2·26-s + 27-s − 3·28-s + 10·29-s + 5·31-s − 32-s − 5·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s + 0.554·13-s + 0.801·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.229·19-s − 0.654·21-s + 1.06·22-s − 1.25·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.566·28-s + 1.85·29-s + 0.898·31-s − 0.176·32-s − 0.870·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.145741418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.145741418\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.810504122961107276296448912027, −8.166360002861459473388767211292, −7.65413840568452465144147907114, −6.61072040183152185812517360312, −6.11532498490045678119089308595, −4.99734969623358779987712591625, −3.84795020439861036846716122060, −2.90043687921487017392042208715, −2.30560140959508307974254558712, −0.70376376823350907184750688404,
0.70376376823350907184750688404, 2.30560140959508307974254558712, 2.90043687921487017392042208715, 3.84795020439861036846716122060, 4.99734969623358779987712591625, 6.11532498490045678119089308595, 6.61072040183152185812517360312, 7.65413840568452465144147907114, 8.166360002861459473388767211292, 8.810504122961107276296448912027
