| L(s) = 1 | − 1.36·2-s − 0.129·4-s − 0.636·5-s + 7-s + 2.91·8-s + 0.870·10-s − 3.59·13-s − 1.36·14-s − 3.72·16-s − 1.46·17-s − 5.85·19-s + 0.0824·20-s − 8.56·23-s − 4.59·25-s + 4.91·26-s − 0.129·28-s − 7.73·29-s − 9.44·31-s − 0.731·32-s + 2·34-s − 0.636·35-s + 5.59·37-s + 8.00·38-s − 1.85·40-s − 8.37·41-s + 1.74·43-s + 11.7·46-s + ⋯ |
| L(s) = 1 | − 0.967·2-s − 0.0647·4-s − 0.284·5-s + 0.377·7-s + 1.02·8-s + 0.275·10-s − 0.997·13-s − 0.365·14-s − 0.931·16-s − 0.354·17-s − 1.34·19-s + 0.0184·20-s − 1.78·23-s − 0.918·25-s + 0.964·26-s − 0.0244·28-s − 1.43·29-s − 1.69·31-s − 0.129·32-s + 0.342·34-s − 0.107·35-s + 0.919·37-s + 1.29·38-s − 0.293·40-s − 1.30·41-s + 0.265·43-s + 1.72·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2782312697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2782312697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 5 | \( 1 + 0.636T + 5T^{2} \) |
| 13 | \( 1 + 3.59T + 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 + 8.56T + 23T^{2} \) |
| 29 | \( 1 + 7.73T + 29T^{2} \) |
| 31 | \( 1 + 9.44T + 31T^{2} \) |
| 37 | \( 1 - 5.59T + 37T^{2} \) |
| 41 | \( 1 + 8.37T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 - 4.99T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 1.85T + 67T^{2} \) |
| 71 | \( 1 + 9.66T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 + 3.08T + 83T^{2} \) |
| 89 | \( 1 + 4.19T + 89T^{2} \) |
| 97 | \( 1 - 7.44T + 97T^{2} \) |
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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
−7.83632191247781064949839365509, −7.55351690431886471455074189059, −6.72235105597460847150545214889, −5.79170905695918210436648474657, −5.05577490046184502203369822429, −4.16690538738535255161732506568, −3.79614801481783340973314293714, −2.16585364846422136126328030950, −1.87168498146448790186960305177, −0.29404026559765735195646324177,
0.29404026559765735195646324177, 1.87168498146448790186960305177, 2.16585364846422136126328030950, 3.79614801481783340973314293714, 4.16690538738535255161732506568, 5.05577490046184502203369822429, 5.79170905695918210436648474657, 6.72235105597460847150545214889, 7.55351690431886471455074189059, 7.83632191247781064949839365509
