| L(s) = 1 | − 0.270·2-s − 3·3-s − 7.92·4-s + 0.812·6-s + 33.4·7-s + 4.31·8-s + 9·9-s + 11·11-s + 23.7·12-s + 33.5·13-s − 9.04·14-s + 62.2·16-s + 71.1·17-s − 2.43·18-s − 48.9·19-s − 100.·21-s − 2.97·22-s − 66.7·23-s − 12.9·24-s − 9.07·26-s − 27·27-s − 264.·28-s − 66.8·29-s + 145.·31-s − 51.3·32-s − 33·33-s − 19.2·34-s + ⋯ |
| L(s) = 1 | − 0.0957·2-s − 0.577·3-s − 0.990·4-s + 0.0552·6-s + 1.80·7-s + 0.190·8-s + 0.333·9-s + 0.301·11-s + 0.572·12-s + 0.715·13-s − 0.172·14-s + 0.972·16-s + 1.01·17-s − 0.0319·18-s − 0.591·19-s − 1.04·21-s − 0.0288·22-s − 0.605·23-s − 0.110·24-s − 0.0684·26-s − 0.192·27-s − 1.78·28-s − 0.428·29-s + 0.844·31-s − 0.283·32-s − 0.174·33-s − 0.0972·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.677924732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677924732\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 0.270T + 8T^{2} \) |
| 7 | \( 1 - 33.4T + 343T^{2} \) |
| 13 | \( 1 - 33.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 66.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 145.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 344.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 34.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 270.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 666.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 876.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 783.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 876.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 523.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 91.0T + 3.89e5T^{2} \) |
| 79 | \( 1 + 96.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 644.T + 9.12e5T^{2} \) |
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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
−9.970532768587816420368011299040, −8.800179492179249001448463962322, −8.236118828426646788282807562506, −7.50498873024003839587616069256, −6.12262739988807942910191058348, −5.26520799439734968431529341567, −4.56935609914032601705235362447, −3.70599297011984617828552456461, −1.75640596820450570718868643980, −0.822792474080395481109250432799,
0.822792474080395481109250432799, 1.75640596820450570718868643980, 3.70599297011984617828552456461, 4.56935609914032601705235362447, 5.26520799439734968431529341567, 6.12262739988807942910191058348, 7.50498873024003839587616069256, 8.236118828426646788282807562506, 8.800179492179249001448463962322, 9.970532768587816420368011299040
