| L(s) = 1 | + (−0.150 + 2.78i)2-s + (−4.15 + 3.11i)3-s + (0.231 + 0.0252i)4-s + (4.42 − 13.1i)5-s + (−8.04 − 12.0i)6-s + (−0.713 − 1.05i)7-s + (−3.71 + 22.6i)8-s + (7.57 − 25.9i)9-s + (35.8 + 14.2i)10-s + (30.4 − 6.69i)11-s + (−1.04 + 0.618i)12-s + (39.4 − 33.5i)13-s + (3.03 − 1.82i)14-s + (22.5 + 68.3i)15-s + (−60.6 − 13.3i)16-s + (83.0 + 56.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.0533 + 0.983i)2-s + (−0.800 + 0.599i)3-s + (0.0289 + 0.00315i)4-s + (0.395 − 1.17i)5-s + (−0.547 − 0.819i)6-s + (−0.0385 − 0.0568i)7-s + (−0.164 + 1.00i)8-s + (0.280 − 0.959i)9-s + (1.13 + 0.452i)10-s + (0.834 − 0.183i)11-s + (−0.0250 + 0.0148i)12-s + (0.842 − 0.715i)13-s + (0.0579 − 0.0348i)14-s + (0.387 + 1.17i)15-s + (−0.947 − 0.208i)16-s + (1.18 + 0.802i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.39542 + 0.891821i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39542 + 0.891821i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.15 - 3.11i)T \) |
| 59 | \( 1 + (69.5 - 447. i)T \) |
| good | 2 | \( 1 + (0.150 - 2.78i)T + (-7.95 - 0.864i)T^{2} \) |
| 5 | \( 1 + (-4.42 + 13.1i)T + (-99.5 - 75.6i)T^{2} \) |
| 7 | \( 1 + (0.713 + 1.05i)T + (-126. + 318. i)T^{2} \) |
| 11 | \( 1 + (-30.4 + 6.69i)T + (1.20e3 - 558. i)T^{2} \) |
| 13 | \( 1 + (-39.4 + 33.5i)T + (355. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-83.0 - 56.2i)T + (1.81e3 + 4.56e3i)T^{2} \) |
| 19 | \( 1 + (-48.6 + 91.7i)T + (-3.84e3 - 5.67e3i)T^{2} \) |
| 23 | \( 1 + (128. - 121. i)T + (658. - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-162. + 8.79i)T + (2.42e4 - 2.63e3i)T^{2} \) |
| 31 | \( 1 + (283. - 150. i)T + (1.67e4 - 2.46e4i)T^{2} \) |
| 37 | \( 1 + (-264. + 43.3i)T + (4.80e4 - 1.61e4i)T^{2} \) |
| 41 | \( 1 + (144. - 152. i)T + (-3.73e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (-101. + 461. i)T + (-7.21e4 - 3.33e4i)T^{2} \) |
| 47 | \( 1 + (-353. + 118. i)T + (8.26e4 - 6.28e4i)T^{2} \) |
| 53 | \( 1 + (-190. + 76.0i)T + (1.08e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-804. - 43.6i)T + (2.25e5 + 2.45e4i)T^{2} \) |
| 67 | \( 1 + (96.4 + 15.8i)T + (2.85e5 + 9.60e4i)T^{2} \) |
| 71 | \( 1 + (-73.5 - 218. i)T + (-2.84e5 + 2.16e5i)T^{2} \) |
| 73 | \( 1 + (-328. - 545. i)T + (-1.82e5 + 3.43e5i)T^{2} \) |
| 79 | \( 1 + (473. + 218. i)T + (3.19e5 + 3.75e5i)T^{2} \) |
| 83 | \( 1 + (-48.0 - 172. i)T + (-4.89e5 + 2.94e5i)T^{2} \) |
| 89 | \( 1 + (47.6 + 878. i)T + (-7.00e5 + 7.62e4i)T^{2} \) |
| 97 | \( 1 + (520. - 865. i)T + (-4.27e5 - 8.06e5i)T^{2} \) |
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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
−12.28288918512761018083679878754, −11.52124521536317651893161881111, −10.35905030271901018839567575671, −9.197981368192617371849677797409, −8.379881092474642005512687327292, −7.00764604175486751161670776252, −5.71325973842966261775869632229, −5.41675436408655374383286056986, −3.80937679703678828891549684488, −1.11879229904186264571162237154,
1.20862979482065196286505876094, 2.46200685607938278017840650244, 3.90709157589838620197022650212, 5.97925970356255578215523599886, 6.59107496962635775460446083931, 7.65928496338007234852311935184, 9.564576023530989336427819127619, 10.34900793207519856524039744303, 11.19347278675664833856478194242, 11.87980269359227767220195048937
