| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s + 6·11-s − 12-s + 4·13-s − 4·14-s + 15-s + 16-s + 6·17-s + 18-s − 20-s + 4·21-s + 6·22-s + 6·23-s − 24-s + 25-s + 4·26-s − 27-s − 4·28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s + 1.10·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.223·20-s + 0.872·21-s + 1.27·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.755·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.779700665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.779700665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−16.31286803719075, −16.17744689422631, −15.39663464320273, −14.73365869380280, −14.38671510425489, −13.39814226296921, −13.14873990697172, −12.46656726380677, −11.95762189767889, −11.46271383804755, −10.95996656470077, −10.05537902870901, −9.623977513229262, −8.956025760850523, −8.181975092094240, −7.222402213725653, −6.669220319867293, −6.315032180732699, −5.685267879764736, −4.876802130821405, −3.888088212230882, −3.574254697840382, −3.002914704114474, −1.502819293810445, −0.7843023436907310,
0.7843023436907310, 1.502819293810445, 3.002914704114474, 3.574254697840382, 3.888088212230882, 4.876802130821405, 5.685267879764736, 6.315032180732699, 6.669220319867293, 7.222402213725653, 8.181975092094240, 8.956025760850523, 9.623977513229262, 10.05537902870901, 10.95996656470077, 11.46271383804755, 11.95762189767889, 12.46656726380677, 13.14873990697172, 13.39814226296921, 14.38671510425489, 14.73365869380280, 15.39663464320273, 16.17744689422631, 16.31286803719075
