| L(s) = 1 | + 1.77·2-s − 4.84·4-s − 6.23·5-s − 18.0·7-s − 22.8·8-s − 11.0·10-s − 13.3·11-s − 66.3·13-s − 32.0·14-s − 1.68·16-s + 97.6·17-s + 109.·19-s + 30.2·20-s − 23.6·22-s + 147.·23-s − 86.1·25-s − 117.·26-s + 87.6·28-s + 173.·29-s + 148.·31-s + 179.·32-s + 173.·34-s + 112.·35-s − 446.·37-s + 194.·38-s + 142.·40-s − 182.·41-s + ⋯ |
| L(s) = 1 | + 0.627·2-s − 0.606·4-s − 0.557·5-s − 0.976·7-s − 1.00·8-s − 0.349·10-s − 0.365·11-s − 1.41·13-s − 0.612·14-s − 0.0263·16-s + 1.39·17-s + 1.32·19-s + 0.337·20-s − 0.229·22-s + 1.33·23-s − 0.689·25-s − 0.888·26-s + 0.591·28-s + 1.11·29-s + 0.861·31-s + 0.991·32-s + 0.874·34-s + 0.543·35-s − 1.98·37-s + 0.831·38-s + 0.561·40-s − 0.696·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.262217989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262217989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 1.77T + 8T^{2} \) |
| 5 | \( 1 + 6.23T + 125T^{2} \) |
| 7 | \( 1 + 18.0T + 343T^{2} \) |
| 11 | \( 1 + 13.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 66.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 97.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 446.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 182.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 223.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 529.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 398.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 788.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 288.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 549.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 190.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 410.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−10.06954740644597920896380562926, −9.804179742545653684968030250921, −8.682991808017173537797694870207, −7.64841538747482001893949858717, −6.78366770378598595476287520961, −5.45050005453073178728837061333, −4.88913238547670250810259248802, −3.52688132693943775866802579484, −2.93823707358891865357203822279, −0.61240722302548474968595519646,
0.61240722302548474968595519646, 2.93823707358891865357203822279, 3.52688132693943775866802579484, 4.88913238547670250810259248802, 5.45050005453073178728837061333, 6.78366770378598595476287520961, 7.64841538747482001893949858717, 8.682991808017173537797694870207, 9.804179742545653684968030250921, 10.06954740644597920896380562926
