| L(s) = 1 | + (0.644 + 0.294i)2-s + (−0.0204 + 0.0697i)3-s + (−2.29 − 2.64i)4-s + (4.83 + 1.28i)5-s + (−0.0337 + 0.0389i)6-s + (1.13 − 7.86i)7-s + (−1.49 − 5.09i)8-s + (7.56 + 4.86i)9-s + (2.73 + 2.24i)10-s + (8.53 − 3.89i)11-s + (0.231 − 0.105i)12-s + (−5.27 + 0.758i)13-s + (3.03 − 4.72i)14-s + (−0.188 + 0.310i)15-s + (−1.45 + 10.1i)16-s + (14.0 − 16.1i)17-s + ⋯ |
| L(s) = 1 | + (0.322 + 0.147i)2-s + (−0.00682 + 0.0232i)3-s + (−0.572 − 0.661i)4-s + (0.966 + 0.256i)5-s + (−0.00561 + 0.00648i)6-s + (0.161 − 1.12i)7-s + (−0.186 − 0.636i)8-s + (0.840 + 0.540i)9-s + (0.273 + 0.224i)10-s + (0.775 − 0.354i)11-s + (0.0192 − 0.00880i)12-s + (−0.405 + 0.0583i)13-s + (0.217 − 0.337i)14-s + (−0.0125 + 0.0207i)15-s + (−0.0910 + 0.633i)16-s + (0.824 − 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.63594 - 0.447389i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63594 - 0.447389i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.83 - 1.28i)T \) |
| 23 | \( 1 + (20.6 - 10.0i)T \) |
| good | 2 | \( 1 + (-0.644 - 0.294i)T + (2.61 + 3.02i)T^{2} \) |
| 3 | \( 1 + (0.0204 - 0.0697i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-1.13 + 7.86i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (-8.53 + 3.89i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (5.27 - 0.758i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (-14.0 + 16.1i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (14.1 - 12.2i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (13.8 - 15.9i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (15.6 - 4.58i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (-9.31 - 5.98i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-21.6 + 13.9i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (53.8 + 15.8i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 - 64.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (8.04 - 55.9i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (2.43 + 16.9i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (-21.6 - 73.8i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-36.8 + 80.5i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-35.2 + 77.2i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (26.3 - 22.7i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-129. + 18.6i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (15.6 + 10.0i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (16.0 - 54.6i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-66.5 + 42.7i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−13.64440686878185609806046997299, −12.51725191856911258707530739162, −10.81869550461435194024634933019, −10.06527420514586320108662995282, −9.317508150877496587440401198023, −7.53419296477106751511761816264, −6.39285023621552930899108466440, −5.15613139408222204155959563283, −3.93609970645113659784745696401, −1.43094529015005892126433853923,
2.12648545516907206357298784324, 3.98141630608555958621251143175, 5.29738952817698506098159556700, 6.52955538760192254360719702012, 8.221250960559658249056645640671, 9.200623746600818344697783210891, 9.980074877974267973532150794137, 11.76466033411746636481336993541, 12.56246847809722507842661333984, 13.10473793749886404775595832099
