| L(s) = 1 | + (15.4 + 1.85i)3-s + 55.8i·5-s + 225. i·7-s + (236. + 57.4i)9-s − 36.9·11-s − 795.·13-s + (−103. + 864. i)15-s + 129. i·17-s + 998. i·19-s + (−419. + 3.49e3i)21-s − 3.74e3·23-s + 4.87·25-s + (3.54e3 + 1.32e3i)27-s + 921. i·29-s + 423. i·31-s + ⋯ |
| L(s) = 1 | + (0.992 + 0.118i)3-s + 0.999i·5-s + 1.74i·7-s + (0.971 + 0.236i)9-s − 0.0920·11-s − 1.30·13-s + (−0.118 + 0.992i)15-s + 0.108i·17-s + 0.634i·19-s + (−0.207 + 1.73i)21-s − 1.47·23-s + 0.00156·25-s + (0.936 + 0.350i)27-s + 0.203i·29-s + 0.0791i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.009570607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009570607\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-15.4 - 1.85i)T \) |
| good | 5 | \( 1 - 55.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 225. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 36.9T + 1.61e5T^{2} \) |
| 13 | \( 1 + 795.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 129. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 998. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.74e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 921. iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 423. iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.44e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 2.08e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.13e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 8.97e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.56e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.15e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.56e5T + 8.58e9T^{2} \) |
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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
−10.77577788460791376709859826195, −9.883893813988508780186996062163, −9.173297236350267791157377957085, −8.208022804957963311100253265719, −7.38795323368471740422672745840, −6.25459958305827728803119081867, −5.17378954326323050852203803512, −3.73677033612078950531508874765, −2.55438074605819484913579520316, −2.14492329229724303729030554178,
0.40787417853871860230899750578, 1.44449634022700964553455699126, 2.86172692549467655646062397988, 4.26598125258466214355532873554, 4.67299614146538367128224951049, 6.49098124539366705539804013001, 7.66812196393900471237599511744, 7.917514143955263557023756394077, 9.327790151766106543832418269391, 9.808904031265550107824521153661
