| L(s) = 1 | + (−15.0 + 4.14i)3-s + (29.4 − 47.5i)5-s + (−98.6 + 98.6i)7-s + (208. − 124. i)9-s − 208. i·11-s + (823. + 823. i)13-s + (−245. + 836. i)15-s + (1.33e3 + 1.33e3i)17-s + 2.16e3i·19-s + (1.07e3 − 1.89e3i)21-s + (2.06e3 − 2.06e3i)23-s + (−1.39e3 − 2.79e3i)25-s + (−2.61e3 + 2.73e3i)27-s − 43.7·29-s + 1.05e3·31-s + ⋯ |
| L(s) = 1 | + (−0.963 + 0.265i)3-s + (0.526 − 0.850i)5-s + (−0.760 + 0.760i)7-s + (0.858 − 0.512i)9-s − 0.518i·11-s + (1.35 + 1.35i)13-s + (−0.281 + 0.959i)15-s + (1.12 + 1.12i)17-s + 1.37i·19-s + (0.530 − 0.935i)21-s + (0.812 − 0.812i)23-s + (−0.446 − 0.894i)25-s + (−0.691 + 0.722i)27-s − 0.00965·29-s + 0.197·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.19324 + 0.457469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19324 + 0.457469i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.0 - 4.14i)T \) |
| 5 | \( 1 + (-29.4 + 47.5i)T \) |
| good | 7 | \( 1 + (98.6 - 98.6i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 208. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-823. - 823. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.33e3 - 1.33e3i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 2.16e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-2.06e3 + 2.06e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 43.7T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-2.19e3 + 2.19e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 7.32e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (5.65e3 + 5.65e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.70e4 - 1.70e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.61e4 - 1.61e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 1.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.89e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-8.17e3 + 8.17e3i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 1.00e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.44e4 - 2.44e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 8.56e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (9.10e3 - 9.10e3i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 3.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (997. - 997. i)T - 8.58e9iT^{2} \) |
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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
−14.11954074729425324889626329483, −12.76484509541826023344332328784, −12.15022381264366259418970604375, −10.78480615033209125987312168646, −9.574706281514976790919064194632, −8.558529421031779756147604342650, −6.29302302106393600943518463450, −5.69314025907126527736744340491, −3.95271242376315976845152286758, −1.30384002838482245781207267966,
0.830441569371714267422289697272, 3.22099672712371521519192306871, 5.32297508140803461229483819187, 6.57281824065611116125843253318, 7.46922339167595625244653000827, 9.702395196565004690599913903291, 10.55560332814855095896968088282, 11.49275433858153508909390939514, 13.07930053118372626130195564362, 13.55858976311862516531721377423
