| L(s) = 1 | + (−0.0336 − 0.103i)3-s + (−4.89 − 3.55i)5-s + (−2.51 − 0.817i)7-s + (7.27 − 5.28i)9-s + (−3.06 + 10.5i)11-s + (2.23 + 3.08i)13-s + (−0.203 + 0.626i)15-s + (−16.3 + 22.4i)17-s + (−34.7 + 11.2i)19-s + 0.288i·21-s − 11.0·23-s + (3.57 + 11.0i)25-s + (−1.58 − 1.15i)27-s + (−19.1 − 6.23i)29-s + (1.75 − 1.27i)31-s + ⋯ |
| L(s) = 1 | + (−0.0112 − 0.0345i)3-s + (−0.978 − 0.710i)5-s + (−0.359 − 0.116i)7-s + (0.807 − 0.587i)9-s + (−0.279 + 0.960i)11-s + (0.172 + 0.237i)13-s + (−0.0135 + 0.0417i)15-s + (−0.958 + 1.31i)17-s + (−1.82 + 0.594i)19-s + 0.0137i·21-s − 0.479·23-s + (0.143 + 0.440i)25-s + (−0.0587 − 0.0426i)27-s + (−0.662 − 0.215i)29-s + (0.0566 − 0.0411i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 - 0.503i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0399634 + 0.148020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0399634 + 0.148020i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.51 + 0.817i)T \) |
| 11 | \( 1 + (3.06 - 10.5i)T \) |
| good | 3 | \( 1 + (0.0336 + 0.103i)T + (-7.28 + 5.29i)T^{2} \) |
| 5 | \( 1 + (4.89 + 3.55i)T + (7.72 + 23.7i)T^{2} \) |
| 13 | \( 1 + (-2.23 - 3.08i)T + (-52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (16.3 - 22.4i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (34.7 - 11.2i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + 11.0T + 529T^{2} \) |
| 29 | \( 1 + (19.1 + 6.23i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-1.75 + 1.27i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-6.09 + 18.7i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (14.6 - 4.77i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + 29.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-14.9 - 45.9i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-20.9 + 15.2i)T + (868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (15.1 - 46.6i)T + (-2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-3.80 + 5.23i)T + (-1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + 84.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-97.5 - 70.8i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (91.6 + 29.7i)T + (4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-23.9 - 32.9i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-0.370 + 0.509i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 12.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-102. + 74.6i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−12.17046747881444838833084268179, −10.88453867360569680842548751875, −10.04550689994533286215487411442, −8.929538287352138520066118127754, −8.118503012268886548861477829493, −7.06529266386715235345023071151, −6.09287599723442233524873968297, −4.26673032270444855748457007395, −4.08616922336717811706278056919, −1.85962475996905096210679163383,
0.07077001452243204205344722897, 2.49230436943766171806439402496, 3.73712394120814198855482999136, 4.83611246106754194186001533669, 6.36084828092708151187669486653, 7.20010163379575028140824880291, 8.136568456637629088005652155742, 9.130316693049847713880673025922, 10.46434986931896839669196546523, 11.00483009844055355127042750434
