| L(s) = 1 | − 5.00·2-s + 17.0·4-s − 3.98·5-s − 45.0·8-s + 19.9·10-s + 5.53·11-s + 33.8·13-s + 89.2·16-s − 103.·17-s + 124.·19-s − 67.7·20-s − 27.6·22-s + 151.·23-s − 109.·25-s − 169.·26-s − 103.·29-s − 60.2·31-s − 85.8·32-s + 516.·34-s + 413.·37-s − 620.·38-s + 179.·40-s − 153.·41-s − 261.·43-s + 94.1·44-s − 755.·46-s + 104.·47-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.12·4-s − 0.356·5-s − 1.99·8-s + 0.629·10-s + 0.151·11-s + 0.721·13-s + 1.39·16-s − 1.47·17-s + 1.49·19-s − 0.757·20-s − 0.268·22-s + 1.36·23-s − 0.873·25-s − 1.27·26-s − 0.663·29-s − 0.349·31-s − 0.474·32-s + 2.60·34-s + 1.83·37-s − 2.64·38-s + 0.709·40-s − 0.585·41-s − 0.927·43-s + 0.322·44-s − 2.42·46-s + 0.325·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7320600174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7320600174\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.00T + 8T^{2} \) |
| 5 | \( 1 + 3.98T + 125T^{2} \) |
| 11 | \( 1 - 5.53T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 151.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 60.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 413.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 153.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 261.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 104.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 716.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 647.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 269.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 483.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 362.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 908.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 162.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 419.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 346.T + 9.12e5T^{2} \) |
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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
−9.426334367768035963867825046876, −8.462455686737349337440975394142, −7.88822654923396762347571682899, −7.03031702041607994550154935313, −6.43755820580178275849361558030, −5.23082777860916198071305303965, −3.85801123059411121913492777103, −2.70888079683286809722394627717, −1.56985479462041078611502291291, −0.58326202738954682314329749742,
0.58326202738954682314329749742, 1.56985479462041078611502291291, 2.70888079683286809722394627717, 3.85801123059411121913492777103, 5.23082777860916198071305303965, 6.43755820580178275849361558030, 7.03031702041607994550154935313, 7.88822654923396762347571682899, 8.462455686737349337440975394142, 9.426334367768035963867825046876
