| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s − 2·9-s − 10-s + 2·11-s − 12-s − 2·13-s + 14-s − 15-s + 16-s − 17-s + 2·18-s + 7·19-s + 20-s + 21-s − 2·22-s − 4·23-s + 24-s − 4·25-s + 2·26-s + 5·27-s − 28-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 − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.471·18-s + 1.60·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.392·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6291609931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6291609931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.64162901829900, −13.18591065413120, −12.16161770827490, −12.02819939794036, −11.84678988866122, −11.08084697941335, −10.65861398416696, −9.992370984209764, −9.803174942509741, −9.238624738264601, −8.707736566385765, −8.300085954326159, −7.479688292723258, −7.220551221873782, −6.548197966001280, −6.055813759572206, −5.657684288616696, −5.134222985615844, −4.494730274755045, −3.600479699203681, −3.183003767562669, −2.461837559847586, −1.819487341891239, −1.140100670620199, −0.3022935232369303,
0.3022935232369303, 1.140100670620199, 1.819487341891239, 2.461837559847586, 3.183003767562669, 3.600479699203681, 4.494730274755045, 5.134222985615844, 5.657684288616696, 6.055813759572206, 6.548197966001280, 7.220551221873782, 7.479688292723258, 8.300085954326159, 8.707736566385765, 9.238624738264601, 9.803174942509741, 9.992370984209764, 10.65861398416696, 11.08084697941335, 11.84678988866122, 12.02819939794036, 12.16161770827490, 13.18591065413120, 13.64162901829900
