| L(s) = 1 | + 4.84·2-s + 3·3-s + 15.4·4-s + 5·5-s + 14.5·6-s + 15.7·7-s + 36.1·8-s + 9·9-s + 24.2·10-s + 26.9·11-s + 46.4·12-s − 46.0·13-s + 76.3·14-s + 15·15-s + 51.5·16-s − 16.2·17-s + 43.6·18-s − 46.2·19-s + 77.3·20-s + 47.2·21-s + 130.·22-s + 120.·23-s + 108.·24-s + 25·25-s − 223.·26-s + 27·27-s + 243.·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.93·4-s + 0.447·5-s + 0.988·6-s + 0.851·7-s + 1.59·8-s + 0.333·9-s + 0.765·10-s + 0.738·11-s + 1.11·12-s − 0.983·13-s + 1.45·14-s + 0.258·15-s + 0.805·16-s − 0.232·17-s + 0.570·18-s − 0.558·19-s + 0.864·20-s + 0.491·21-s + 1.26·22-s + 1.09·23-s + 0.923·24-s + 0.200·25-s − 1.68·26-s + 0.192·27-s + 1.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.03375264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.03375264\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 163 | \( 1 + 163T \) |
| good | 2 | \( 1 - 4.84T + 8T^{2} \) |
| 7 | \( 1 - 15.7T + 343T^{2} \) |
| 11 | \( 1 - 26.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 252.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 118.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 56.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 454.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 278.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 142.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 367.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 91.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 144.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 306.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 775.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 906.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 590.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 817.T + 9.12e5T^{2} \) |
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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
−8.551229002344588438727816244273, −7.56133796679692227375384625748, −6.80775083902974204577750731686, −6.19869979702178638804256985807, −5.11561902382474182783634189940, −4.66869394560596935073553845793, −3.93315057585913746110361652117, −2.82081672458144282225423852136, −2.28634871059278639527529408716, −1.19954994544531687835709543076,
1.19954994544531687835709543076, 2.28634871059278639527529408716, 2.82081672458144282225423852136, 3.93315057585913746110361652117, 4.66869394560596935073553845793, 5.11561902382474182783634189940, 6.19869979702178638804256985807, 6.80775083902974204577750731686, 7.56133796679692227375384625748, 8.551229002344588438727816244273
