| L(s) = 1 | + (−0.142 − 0.237i)2-s + (2.84 + 0.963i)3-s + (1.83 − 3.46i)4-s + (−0.186 − 1.71i)5-s + (−0.177 − 0.812i)6-s + (5.16 − 1.73i)7-s + (−2.19 + 0.118i)8-s + (7.14 + 5.47i)9-s + (−0.380 + 0.289i)10-s + (−14.2 − 9.65i)11-s + (8.55 − 8.07i)12-s + (5.25 + 4.97i)13-s + (−1.15 − 0.977i)14-s + (1.12 − 5.04i)15-s + (−8.46 − 12.4i)16-s + (1.38 − 4.10i)17-s + ⋯ |
| L(s) = 1 | + (−0.0714 − 0.118i)2-s + (0.947 + 0.321i)3-s + (0.459 − 0.866i)4-s + (−0.0372 − 0.342i)5-s + (−0.0295 − 0.135i)6-s + (0.737 − 0.248i)7-s + (−0.274 + 0.0148i)8-s + (0.793 + 0.608i)9-s + (−0.0380 + 0.0289i)10-s + (−1.29 − 0.878i)11-s + (0.713 − 0.673i)12-s + (0.404 + 0.383i)13-s + (−0.0821 − 0.0698i)14-s + (0.0747 − 0.336i)15-s + (−0.529 − 0.780i)16-s + (0.0813 − 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.00761 - 0.771430i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00761 - 0.771430i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.84 - 0.963i)T \) |
| 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
−12.37724215017343809065412137904, −10.91315394690590838106779281056, −10.54230463955499076932507786757, −9.301109914138898056002221627110, −8.337598115173842331075812459491, −7.40004745243401913165959490370, −5.75994298574033387331964477946, −4.67524981500253228057574952766, −3.00136519751188966926382127526, −1.47380730366320626296138352080,
2.19374483340798072955921342758, 3.18842064629945410980126077040, 4.79473859166187928803924246410, 6.66417192001375661068764055761, 7.66743273704392974093254832488, 8.170870665747861261740487738562, 9.329935342597618681889194862565, 10.67431328339982280559062137321, 11.64478414040282831880048847898, 12.88318982842071418987755513580
