| L(s) = 1 | − 2·2-s − 9·3-s − 28·4-s + 46·5-s + 18·6-s + 148·7-s + 120·8-s + 81·9-s − 92·10-s + 121·11-s + 252·12-s + 574·13-s − 296·14-s − 414·15-s + 656·16-s − 722·17-s − 162·18-s + 2.16e3·19-s − 1.28e3·20-s − 1.33e3·21-s − 242·22-s − 2.53e3·23-s − 1.08e3·24-s − 1.00e3·25-s − 1.14e3·26-s − 729·27-s − 4.14e3·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s + 0.822·5-s + 0.204·6-s + 1.14·7-s + 0.662·8-s + 1/3·9-s − 0.290·10-s + 0.301·11-s + 0.505·12-s + 0.942·13-s − 0.403·14-s − 0.475·15-s + 0.640·16-s − 0.605·17-s − 0.117·18-s + 1.37·19-s − 0.720·20-s − 0.659·21-s − 0.106·22-s − 0.999·23-s − 0.382·24-s − 0.322·25-s − 0.333·26-s − 0.192·27-s − 0.998·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.176604191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.176604191\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + p T + p^{5} T^{2} \) |
| 5 | \( 1 - 46 T + p^{5} T^{2} \) |
| 7 | \( 1 - 148 T + p^{5} T^{2} \) |
| 13 | \( 1 - 574 T + p^{5} T^{2} \) |
| 17 | \( 1 + 722 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2160 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2536 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4650 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5032 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8118 T + p^{5} T^{2} \) |
| 41 | \( 1 + 5138 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8304 T + p^{5} T^{2} \) |
| 47 | \( 1 - 24728 T + p^{5} T^{2} \) |
| 53 | \( 1 + 28746 T + p^{5} T^{2} \) |
| 59 | \( 1 + 5860 T + p^{5} T^{2} \) |
| 61 | \( 1 + 53658 T + p^{5} T^{2} \) |
| 67 | \( 1 - 30908 T + p^{5} T^{2} \) |
| 71 | \( 1 + 69648 T + p^{5} T^{2} \) |
| 73 | \( 1 + 18446 T + p^{5} T^{2} \) |
| 79 | \( 1 + 25300 T + p^{5} T^{2} \) |
| 83 | \( 1 + 17556 T + p^{5} T^{2} \) |
| 89 | \( 1 - 132570 T + p^{5} T^{2} \) |
| 97 | \( 1 - 70658 T + p^{5} T^{2} \) |
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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
−15.90030767269105705842286664249, −14.15947957636882431597188319922, −13.52218881054187531776200162044, −11.85003592538449612068029697553, −10.53751455228906256242226959762, −9.313272384197628648374815930076, −7.971399737162129553526268175387, −5.91031575310589564754799983038, −4.52220299211329893679080348766, −1.27590322593231934581374699241,
1.27590322593231934581374699241, 4.52220299211329893679080348766, 5.91031575310589564754799983038, 7.971399737162129553526268175387, 9.313272384197628648374815930076, 10.53751455228906256242226959762, 11.85003592538449612068029697553, 13.52218881054187531776200162044, 14.15947957636882431597188319922, 15.90030767269105705842286664249
