| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s + 11-s − 12-s + 4·14-s + 16-s + 17-s + 18-s + 2·19-s − 4·21-s + 22-s − 2·23-s − 24-s − 5·25-s − 27-s + 4·28-s + 6·29-s + 8·31-s + 32-s − 33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.872·21-s + 0.213·22-s − 0.417·23-s − 0.204·24-s − 25-s − 0.192·27-s + 0.755·28-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.808460084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.808460084\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.06002805181947, −12.58854111951012, −11.98034422704853, −11.75044124337114, −11.47447775071359, −10.87212922063283, −10.44627347804306, −9.943642310607082, −9.462537617028144, −8.690140742309173, −8.176027270033989, −7.817517949030372, −7.381547294455013, −6.663598934033301, −6.229924168864202, −5.792322921402719, −5.075068513181853, −4.872694746959943, −4.323100121838043, −3.857652488547443, −3.085758977305503, −2.504690002864019, −1.660753643552355, −1.418472307575732, −0.5756125945085334,
0.5756125945085334, 1.418472307575732, 1.660753643552355, 2.504690002864019, 3.085758977305503, 3.857652488547443, 4.323100121838043, 4.872694746959943, 5.075068513181853, 5.792322921402719, 6.229924168864202, 6.663598934033301, 7.381547294455013, 7.817517949030372, 8.176027270033989, 8.690140742309173, 9.462537617028144, 9.943642310607082, 10.44627347804306, 10.87212922063283, 11.47447775071359, 11.75044124337114, 11.98034422704853, 12.58854111951012, 13.06002805181947
