| L(s) = 1 | + 4.89·2-s + 15.9·4-s − 15.5·5-s − 7·7-s + 38.7·8-s − 76.0·10-s + 11·11-s + 74.3·13-s − 34.2·14-s + 62.1·16-s + 94.0·17-s + 135.·19-s − 247.·20-s + 53.8·22-s − 81.1·23-s + 116.·25-s + 363.·26-s − 111.·28-s + 53.4·29-s − 9.50·31-s − 6.09·32-s + 460.·34-s + 108.·35-s − 9.14·37-s + 663.·38-s − 602.·40-s + 339.·41-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 1.99·4-s − 1.39·5-s − 0.377·7-s + 1.71·8-s − 2.40·10-s + 0.301·11-s + 1.58·13-s − 0.653·14-s + 0.970·16-s + 1.34·17-s + 1.63·19-s − 2.76·20-s + 0.521·22-s − 0.735·23-s + 0.934·25-s + 2.74·26-s − 0.752·28-s + 0.342·29-s − 0.0550·31-s − 0.0336·32-s + 2.32·34-s + 0.525·35-s − 0.0406·37-s + 2.83·38-s − 2.38·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.039504338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.039504338\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 4.89T + 8T^{2} \) |
| 5 | \( 1 + 15.5T + 125T^{2} \) |
| 13 | \( 1 - 74.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 94.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 81.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 53.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.50T + 2.97e4T^{2} \) |
| 37 | \( 1 + 9.14T + 5.06e4T^{2} \) |
| 41 | \( 1 - 339.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 433.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 54.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 534.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 358.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 694.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 278.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 886.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 185.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 122.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 847.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−10.49159558754051755333207514492, −9.165511259575618929252079631984, −7.910208531091017473926810567349, −7.32462406222324063517239359595, −6.17744083731409831271354024955, −5.52478209920257805696446722937, −4.25805424214721117983727123507, −3.66228378482822869159315235371, −3.00243769870138616923756200921, −1.07463519696312369015890381294,
1.07463519696312369015890381294, 3.00243769870138616923756200921, 3.66228378482822869159315235371, 4.25805424214721117983727123507, 5.52478209920257805696446722937, 6.17744083731409831271354024955, 7.32462406222324063517239359595, 7.910208531091017473926810567349, 9.165511259575618929252079631984, 10.49159558754051755333207514492
