| L(s) = 1 | + (2.87 + 0.873i)3-s − 2.23i·5-s + 11.2·7-s + (7.47 + 5.01i)9-s − 16.7i·11-s − 0.328·13-s + (1.95 − 6.41i)15-s − 9.16i·17-s − 18.6·19-s + (32.1 + 9.78i)21-s + 19.3i·23-s − 5.00·25-s + (17.0 + 20.9i)27-s − 9.30i·29-s − 9.75·31-s + ⋯ |
| L(s) = 1 | + (0.956 + 0.291i)3-s − 0.447i·5-s + 1.60·7-s + (0.830 + 0.556i)9-s − 1.52i·11-s − 0.0252·13-s + (0.130 − 0.427i)15-s − 0.539i·17-s − 0.982·19-s + (1.53 + 0.465i)21-s + 0.843i·23-s − 0.200·25-s + (0.632 + 0.774i)27-s − 0.320i·29-s − 0.314·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.291i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.874911792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.874911792\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.87 - 0.873i)T \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 - 11.2T + 49T^{2} \) |
| 11 | \( 1 + 16.7iT - 121T^{2} \) |
| 13 | \( 1 + 0.328T + 169T^{2} \) |
| 17 | \( 1 + 9.16iT - 289T^{2} \) |
| 19 | \( 1 + 18.6T + 361T^{2} \) |
| 23 | \( 1 - 19.3iT - 529T^{2} \) |
| 29 | \( 1 + 9.30iT - 841T^{2} \) |
| 31 | \( 1 + 9.75T + 961T^{2} \) |
| 37 | \( 1 - 71.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 23.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 9.32T + 1.84e3T^{2} \) |
| 47 | \( 1 - 46.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 84.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 76.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.17T + 3.72e3T^{2} \) |
| 67 | \( 1 + 97.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 94.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 123.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 85.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 49.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 114. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 113.T + 9.40e3T^{2} \) |
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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
−10.97192350899440957678457017347, −9.649548465865145241473825505030, −8.757742557125473595892506892905, −8.189673336352973977684088915899, −7.50073308224829852352093821845, −5.88821533146490051500545705915, −4.83218901593088655823803981275, −3.95319593536240268373253858960, −2.57908886954554429102594569291, −1.25514776856039649842666534454,
1.65217457493286558007030554060, 2.43091516083105912710639426397, 4.08751105231095684689732732585, 4.80120868775902874803007452786, 6.43022433397279969606878744893, 7.40102045632704790243761951713, 8.051976424210194335832336774083, 8.850303284339150121377844915369, 9.957774959955875932576403958802, 10.71569690105887104665951827962
