| L(s) = 1 | + (−4.71 + 1.53i)3-s + (3.20 + 10.7i)5-s − 16.8i·7-s + (−1.97 + 1.43i)9-s + (−47.8 − 34.7i)11-s + (−3.71 − 5.10i)13-s + (−31.4 − 45.5i)15-s + (−125. − 40.6i)17-s + (−25.3 + 78.1i)19-s + (25.8 + 79.4i)21-s + (93.7 − 128. i)23-s + (−104. + 68.5i)25-s + (85.7 − 118. i)27-s + (15.4 + 47.5i)29-s + (−13.8 + 42.4i)31-s + ⋯ |
| L(s) = 1 | + (−0.907 + 0.294i)3-s + (0.286 + 0.958i)5-s − 0.910i·7-s + (−0.0730 + 0.0530i)9-s + (−1.31 − 0.952i)11-s + (−0.0791 − 0.108i)13-s + (−0.542 − 0.784i)15-s + (−1.78 − 0.580i)17-s + (−0.306 + 0.943i)19-s + (0.268 + 0.825i)21-s + (0.849 − 1.16i)23-s + (−0.835 + 0.548i)25-s + (0.611 − 0.841i)27-s + (0.0990 + 0.304i)29-s + (−0.0799 + 0.246i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0115194 - 0.0497314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0115194 - 0.0497314i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-3.20 - 10.7i)T \) |
| good | 3 | \( 1 + (4.71 - 1.53i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 + 16.8iT - 343T^{2} \) |
| 11 | \( 1 + (47.8 + 34.7i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (3.71 + 5.10i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (125. + 40.6i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (25.3 - 78.1i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-93.7 + 128. i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-15.4 - 47.5i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (13.8 - 42.4i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-44.1 - 60.7i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (286. - 207. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 79.3iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (289. - 94.1i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-322. + 104. i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (636. - 462. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-675. - 490. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (578. + 187. i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (18.9 + 58.4i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (23.2 - 32.0i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (178. + 549. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-50.7 - 16.4i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (1.17e3 + 850. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-50.9 + 16.5i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−13.13056892243332604820804582064, −11.46119585479468856341366664525, −10.72124787761818681807026778842, −10.28246736747764445142547070663, −8.435820024060410372823188466216, −7.02855293337905365425480153686, −5.99287156647890505667136255390, −4.67976923063420912110908483564, −2.83218985552868060073472182185, −0.02992021358528851905187808200,
2.16607348975135826614431070104, 4.79717078070951158590249805420, 5.58423563497763455297796271039, 6.88375915956341049119361101288, 8.488195337514933460566609956790, 9.404322228361807569695880395365, 10.85843694058105157098805864739, 11.83769718693123190500864148650, 12.80198860625295830907394693387, 13.32399904338076264481311678775
