| L(s) = 1 | + (−5.70 + 6.95i)3-s − 34.7i·5-s + 89.5·7-s + (−15.8 − 79.4i)9-s + 131. i·11-s + 174.·13-s + (241. + 198. i)15-s − 221. i·17-s − 317.·19-s + (−511. + 623. i)21-s − 322. i·23-s − 579.·25-s + (643. + 343. i)27-s − 365. i·29-s + 1.13e3·31-s + ⋯ |
| L(s) = 1 | + (−0.634 + 0.773i)3-s − 1.38i·5-s + 1.82·7-s + (−0.195 − 0.980i)9-s + 1.08i·11-s + 1.03·13-s + (1.07 + 0.880i)15-s − 0.766i·17-s − 0.878·19-s + (−1.15 + 1.41i)21-s − 0.609i·23-s − 0.927·25-s + (0.882 + 0.470i)27-s − 0.434i·29-s + 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.983920609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.983920609\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.70 - 6.95i)T \) |
| good | 5 | \( 1 + 34.7iT - 625T^{2} \) |
| 7 | \( 1 - 89.5T + 2.40e3T^{2} \) |
| 11 | \( 1 - 131. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 174.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 221. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 317.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 322. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 365. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.13e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.99e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.96e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.54e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.11e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.26e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 178. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.29e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 6.53e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 69.7iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.21e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.07e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.98e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.27e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 5.22e3T + 8.85e7T^{2} \) |
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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
−10.67893110237513587634525815509, −9.740540033814479429088094975004, −8.609881774292764345307668162545, −8.255241286354087243535954098865, −6.69803596801973883674904593786, −5.26936109163961071963046129980, −4.81226827590293743941865717168, −4.09779979386163127749593487316, −1.83712025989212847693425660366, −0.72461638855893701865403901419,
1.16610385126601430352165082797, 2.23051855880677100583695046134, 3.71992316890933757001248597159, 5.20400263852961812142986176657, 6.14611398785174399900202583420, 6.92950130753150412070381314038, 8.079708585972457805350926990838, 8.480443444606374085779888665163, 10.51978402021111611342925143223, 10.98100310155387980572807059306
