| L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s + 7-s + 8-s + 9-s + 4·10-s + 4·11-s + 12-s − 13-s + 14-s + 4·15-s + 16-s − 17-s + 18-s + 4·20-s + 21-s + 4·22-s + 4·23-s + 24-s + 11·25-s − 26-s + 27-s + 28-s + 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.894·20-s + 0.218·21-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.175784545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.175784545\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.43431622225293476980256076857, −6.91485885108834120445811403430, −6.30002641476588263934450780620, −5.58752905368971229950122030594, −5.05216499506929163214392185002, −4.23639732023900214253770110943, −3.39905678903158564906056117992, −2.57135603277719859180202295969, −1.83970930829277438866929754733, −1.29631681266387487357538053133,
1.29631681266387487357538053133, 1.83970930829277438866929754733, 2.57135603277719859180202295969, 3.39905678903158564906056117992, 4.23639732023900214253770110943, 5.05216499506929163214392185002, 5.58752905368971229950122030594, 6.30002641476588263934450780620, 6.91485885108834120445811403430, 7.43431622225293476980256076857
