| L(s) = 1 | + (−1.37 + 1.45i)2-s + (−2.01 + 1.02i)3-s + (−0.211 − 3.99i)4-s + (4.19 − 2.72i)5-s + (1.28 − 4.34i)6-s + (−2.63 + 5.16i)7-s + (6.08 + 5.19i)8-s + (−2.27 + 3.12i)9-s + (−1.81 + 9.83i)10-s + (−9.84 − 4.90i)11-s + (4.53 + 7.84i)12-s + (−2.39 − 15.1i)13-s + (−3.87 − 10.9i)14-s + (−5.65 + 9.81i)15-s + (−15.9 + 1.68i)16-s + (−1.91 + 12.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.688 + 0.725i)2-s + (−0.672 + 0.342i)3-s + (−0.0528 − 0.998i)4-s + (0.838 − 0.545i)5-s + (0.214 − 0.724i)6-s + (−0.375 + 0.737i)7-s + (0.760 + 0.648i)8-s + (−0.252 + 0.347i)9-s + (−0.181 + 0.983i)10-s + (−0.894 − 0.446i)11-s + (0.377 + 0.653i)12-s + (−0.184 − 1.16i)13-s + (−0.276 − 0.780i)14-s + (−0.377 + 0.654i)15-s + (−0.994 + 0.105i)16-s + (−0.112 + 0.711i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.106567 - 0.143223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106567 - 0.143223i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 1.45i)T \) |
| 5 | \( 1 + (-4.19 + 2.72i)T \) |
| 11 | \( 1 + (9.84 + 4.90i)T \) |
| good | 3 | \( 1 + (2.01 - 1.02i)T + (5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (2.63 - 5.16i)T + (-28.8 - 39.6i)T^{2} \) |
| 13 | \( 1 + (2.39 + 15.1i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (1.91 - 12.0i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (24.2 + 7.89i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (-11.4 - 11.4i)T + 529iT^{2} \) |
| 29 | \( 1 + (16.0 + 49.3i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-7.84 + 10.7i)T + (-296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-4.00 + 7.85i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (75.7 + 24.6i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (54.5 - 54.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (22.4 + 44.0i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (21.8 - 3.46i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (15.0 + 46.4i)T + (-2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-55.4 - 76.2i)T + (-1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (5.11 - 5.11i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + (-10.7 - 14.8i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (46.5 - 91.4i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-47.7 + 65.7i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-9.70 + 61.2i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + 11.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-7.98 - 50.3i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−11.49047283335376805278662063916, −10.44917808341461167024310155265, −9.927926271850697573529415747056, −8.675309090291551259935430384848, −8.048192348411691232829240772162, −6.30250651099162112766332402069, −5.65627136871489161750436152252, −4.93407269160207678525579521098, −2.32938262563847444606507132919, −0.12481792012100827891923159346,
1.79275577190846907475689636509, 3.24740609759028949517711807274, 4.92651108274901195019172181115, 6.67882939830806399650125353834, 7.00628840599711898371853004105, 8.642667683411815710789318997545, 9.674454249669540299447432230168, 10.49081376579710622179399124729, 11.15667591845134437994718110353, 12.24387414421737666653141980650
