| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 6.74·13-s + 14-s − 15-s + 16-s − 6.74·17-s − 18-s + 20-s + 21-s + 22-s + 4.74·23-s + 24-s + 25-s − 6.74·26-s − 27-s − 28-s + 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 1.87·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s − 1.63·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.989·23-s + 0.204·24-s + 0.200·25-s − 1.32·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.069099617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.069099617\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 6.74T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2.74T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 6.74T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.956778234347635442730637104321, −8.487979901839505509216166241780, −7.41142902208161687617869934174, −6.48531343587352768607654740423, −6.22396805070377232395179518037, −5.20915611984337887454528193985, −4.16189555862417888490124038059, −3.07039281768085210392715996061, −1.91368269068418361600292607400, −0.77512895009577361479327765091,
0.77512895009577361479327765091, 1.91368269068418361600292607400, 3.07039281768085210392715996061, 4.16189555862417888490124038059, 5.20915611984337887454528193985, 6.22396805070377232395179518037, 6.48531343587352768607654740423, 7.41142902208161687617869934174, 8.487979901839505509216166241780, 8.956778234347635442730637104321
