L(s) = 1 | − 5.19·3-s − 31.1i·7-s + 27·9-s + 70i·13-s + 155.·19-s + 162i·21-s − 125·25-s − 140.·27-s + 155. i·31-s − 110i·37-s − 363. i·39-s + 218.·43-s − 629·49-s − 810·57-s + 182i·61-s + ⋯ |
L(s) = 1 | − 1.00·3-s − 1.68i·7-s + 9-s + 1.49i·13-s + 1.88·19-s + 1.68i·21-s − 25-s − 1.00·27-s + 0.903i·31-s − 0.488i·37-s − 1.49i·39-s + 0.773·43-s − 1.83·49-s − 1.88·57-s + 0.382i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.362973739\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.362973739\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19T \) |
good | 5 | \( 1 + 125T^{2} \) |
| 7 | \( 1 + 31.1iT - 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 - 70iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 155. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 110iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 - 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 - 2.05e5T^{2} \) |
| 61 | \( 1 - 182iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−9.925950123392018401974744948104, −9.289799993430142439616095631090, −7.71603898980661197534846933213, −7.16467530793755123608534981311, −6.44854651293924181005811722430, −5.28635984110818305702164819532, −4.36144709228684008276385914395, −3.62047601524181520318039335250, −1.62137991434244596865235610125, −0.63050624707714154825009934729,
0.817882817300581082198919422250, 2.31416284767667679521119821656, 3.49052771749545545092349896134, 5.06377479949282502051850852729, 5.55893651135371698107494499026, 6.19682136607361985120193838625, 7.49186686124235496714134414860, 8.216285344807150643698356668974, 9.457723474860444949751843307843, 9.906629284311014729182702365414