| L(s) = 1 | − 28.9i·3-s + (13.1 + 54.3i)5-s − 146. i·7-s − 594.·9-s + 191.·11-s + 83.9i·13-s + (1.57e3 − 380. i)15-s − 2.00e3i·17-s − 677.·19-s − 4.24e3·21-s + 1.29e3i·23-s + (−2.77e3 + 1.42e3i)25-s + 1.01e4i·27-s − 3.26e3·29-s − 6.15e3·31-s + ⋯ |
| L(s) = 1 | − 1.85i·3-s + (0.235 + 0.971i)5-s − 1.13i·7-s − 2.44·9-s + 0.476·11-s + 0.137i·13-s + (1.80 − 0.436i)15-s − 1.67i·17-s − 0.430·19-s − 2.10·21-s + 0.510i·23-s + (−0.889 + 0.457i)25-s + 2.68i·27-s − 0.721·29-s − 1.15·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4189892309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4189892309\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-13.1 - 54.3i)T \) |
| good | 3 | \( 1 + 28.9iT - 243T^{2} \) |
| 7 | \( 1 + 146. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 191.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 83.9iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.00e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 677.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.29e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.29e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 9.52e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.47e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.58e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.17e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.33e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.33e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.08e4iT - 8.58e9T^{2} \) |
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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
−10.29177022311435080114285053437, −9.056513629210543630622162000746, −7.76540231531110527103900515912, −7.04946615737477470456236501245, −6.72938189365996536497498917980, −5.47364256537884308102160581336, −3.58013914667909384033123606220, −2.39931365015476864714155616922, −1.29449181091983282571858998923, −0.10764378688552703728341938468,
2.02336723147824760647880790916, 3.58009702018744692993280109330, 4.42547569249193846963346775934, 5.45704575841355437778465569489, 6.03102325243435637194464403821, 8.281470520347435957699838204114, 8.874161293533770002945730191334, 9.468913936988055594416295332596, 10.41169753339392515834328403498, 11.26603559959797810731928503365
