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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.615601 + 0.484302i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.615601 + 0.484302i\) |
\(L(\frac{7}{2})\) |
|
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
−13.50621516504455444530534697512, −12.47412388127935703308965918199, −11.82073877542376439838245345382, −10.62631365672036473990514226538, −8.666711888740953578949656065249, −7.78311528427012704753213396821, −6.62705013023370737733752492544, −5.78743748465514542757407082679, −2.95768738128760157675755817947, −1.81151780013200816661634208893,
0.32147471482301591902389277579, 3.39038826658155224344875657145, 4.55491566531707498269448921742, 5.34370140244901694473813929834, 7.61427895334027185491788608330, 9.008720720064886483975209784788, 9.795208307308192856169786256322, 10.79322143357093212535059828276, 11.77641425345443417251652814258, 13.40973370879862488971101208876