| L(s) = 1 | + (0.142 + 0.237i)2-s + (−1.40 + 2.65i)3-s + (1.83 − 3.46i)4-s + (0.186 + 1.71i)5-s + (−0.830 + 0.0465i)6-s + (5.16 − 1.73i)7-s + (2.19 − 0.118i)8-s + (−5.07 − 7.43i)9-s + (−0.380 + 0.289i)10-s + (14.2 + 9.65i)11-s + (6.62 + 9.72i)12-s + (5.25 + 4.97i)13-s + (1.15 + 0.977i)14-s + (−4.80 − 1.90i)15-s + (−8.46 − 12.4i)16-s + (−1.38 + 4.10i)17-s + ⋯ |
| L(s) = 1 | + (0.0714 + 0.118i)2-s + (−0.466 + 0.884i)3-s + (0.459 − 0.866i)4-s + (0.0372 + 0.342i)5-s + (−0.138 + 0.00776i)6-s + (0.737 − 0.248i)7-s + (0.274 − 0.0148i)8-s + (−0.564 − 0.825i)9-s + (−0.0380 + 0.0289i)10-s + (1.29 + 0.878i)11-s + (0.551 + 0.810i)12-s + (0.404 + 0.383i)13-s + (0.0821 + 0.0698i)14-s + (−0.320 − 0.127i)15-s + (−0.529 − 0.780i)16-s + (−0.0813 + 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.56476 + 0.552074i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56476 + 0.552074i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.40 - 2.65i)T \) |
| 59 | \( 1 + (-4.70 + 58.8i)T \) |
| good | 2 | \( 1 + (-0.142 - 0.237i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (-0.186 - 1.71i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (-5.16 + 1.73i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (-14.2 - 9.65i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (-5.25 - 4.97i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (1.38 - 4.10i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (-4.02 - 24.5i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (4.61 - 1.28i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (-7.81 + 12.9i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (3.10 - 18.9i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-1.03 + 19.1i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (38.2 + 10.6i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (28.3 + 41.7i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (-3.04 + 27.9i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-45.5 + 59.8i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (89.9 - 54.1i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (1.81 + 33.4i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (7.01 - 64.5i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (43.7 - 51.4i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (-19.1 + 47.9i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (50.6 + 109. i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (43.3 - 71.9i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (-39.3 - 46.3i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−12.12445739753734178241790919860, −11.47697299348253843721782362057, −10.50311554847636207657639520476, −9.891658099360291081793548446871, −8.722918561252344226031995796967, −7.03214792582406189767854614264, −6.16395588467573966273385756736, −4.97379809778666862961357176291, −3.87356550450946236664066421992, −1.59048329254120833645448805701,
1.31483621505887120502324641731, 2.98968350276813255292636446643, 4.72607565957397845888675040595, 6.16006409394644794744704027292, 7.10571370470190743511535509672, 8.241484224182739242253400900910, 8.920634698736189110237929586460, 10.97044450400841049730769832171, 11.49113310187445417628478516325, 12.20760770928767995324586802068
