L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 18.2·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 36.5·10-s − 49.0·11-s − 12·12-s − 87.3·13-s − 14·14-s + 54.8·15-s + 16·16-s + 60.5·17-s + 18·18-s − 124.·19-s − 73.1·20-s + 21·21-s − 98.0·22-s + 23·23-s − 24·24-s + 209.·25-s − 174.·26-s − 27·27-s − 28·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.63·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.15·10-s − 1.34·11-s − 0.288·12-s − 1.86·13-s − 0.267·14-s + 0.943·15-s + 0.250·16-s + 0.864·17-s + 0.235·18-s − 1.50·19-s − 0.817·20-s + 0.218·21-s − 0.950·22-s + 0.208·23-s − 0.204·24-s + 1.67·25-s − 1.31·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6512763820\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6512763820\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
good | 5 | \( 1 + 18.2T + 125T^{2} \) |
| 11 | \( 1 + 49.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 124.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 87.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 296.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 174.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 47.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 489.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 152.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 185.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 10.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 680.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 898.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 283.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 618.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 336.T + 9.12e5T^{2} \) |
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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
−10.06741168276335474637515206465, −8.538070019252671101973667857838, −7.60820623517762665688754626894, −7.26172045685715203619171578517, −6.13470701473656430136511088254, −4.96304836166834251747489663419, −4.54368326268435964805623320797, −3.39260067678410510459112624678, −2.44081615697129816847909284853, −0.37428751510762374687025625249,
0.37428751510762374687025625249, 2.44081615697129816847909284853, 3.39260067678410510459112624678, 4.54368326268435964805623320797, 4.96304836166834251747489663419, 6.13470701473656430136511088254, 7.26172045685715203619171578517, 7.60820623517762665688754626894, 8.538070019252671101973667857838, 10.06741168276335474637515206465