| L(s) = 1 | + 5.24·2-s + 19.4·4-s − 4.06·5-s − 7·7-s + 60.2·8-s − 21.2·10-s − 15.1·11-s + 48.5·13-s − 36.7·14-s + 160.·16-s + 107.·17-s + 109.·19-s − 79.1·20-s − 79.4·22-s + 184.·23-s − 108.·25-s + 254.·26-s − 136.·28-s + 94.0·29-s + 135.·31-s + 357.·32-s + 562.·34-s + 28.4·35-s − 149.·37-s + 575.·38-s − 244.·40-s − 297.·41-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.43·4-s − 0.363·5-s − 0.377·7-s + 2.66·8-s − 0.673·10-s − 0.415·11-s + 1.03·13-s − 0.700·14-s + 2.50·16-s + 1.53·17-s + 1.32·19-s − 0.885·20-s − 0.769·22-s + 1.67·23-s − 0.868·25-s + 1.92·26-s − 0.921·28-s + 0.602·29-s + 0.783·31-s + 1.97·32-s + 2.83·34-s + 0.137·35-s − 0.665·37-s + 2.45·38-s − 0.967·40-s − 1.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.465832710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.465832710\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 5.24T + 8T^{2} \) |
| 5 | \( 1 + 4.06T + 125T^{2} \) |
| 11 | \( 1 + 15.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 184.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 94.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 386.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 73.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 633.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 325.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 69.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 279.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 497.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 457.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 78.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 292.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 164.T + 9.12e5T^{2} \) |
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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
−10.70243422075016381992727479837, −9.754261388550596626118377994066, −8.229651663062471400048724304486, −7.31688119544026149393270394562, −6.42937548884364016621065520706, −5.50062252063801476637424758584, −4.79524469064234504387469156878, −3.39938198854248527955597527184, −3.15455349505533992262575686249, −1.34623350256751056248773129118,
1.34623350256751056248773129118, 3.15455349505533992262575686249, 3.39938198854248527955597527184, 4.79524469064234504387469156878, 5.50062252063801476637424758584, 6.42937548884364016621065520706, 7.31688119544026149393270394562, 8.229651663062471400048724304486, 9.754261388550596626118377994066, 10.70243422075016381992727479837
