| L(s) = 1 | + (−0.212 + 3.91i)2-s + (4.58 + 2.43i)3-s + (−7.33 − 0.798i)4-s + (−0.421 + 1.25i)5-s + (−10.5 + 17.4i)6-s + (−5.56 − 8.21i)7-s + (−0.391 + 2.38i)8-s + (15.1 + 22.3i)9-s + (−4.80 − 1.91i)10-s + (−29.8 + 6.57i)11-s + (−31.7 − 21.5i)12-s + (−45.8 + 38.9i)13-s + (33.3 − 20.0i)14-s + (−4.98 + 4.71i)15-s + (−66.9 − 14.7i)16-s + (23.9 + 16.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.0750 + 1.38i)2-s + (0.883 + 0.469i)3-s + (−0.917 − 0.0997i)4-s + (−0.0376 + 0.111i)5-s + (−0.715 + 1.18i)6-s + (−0.300 − 0.443i)7-s + (−0.0173 + 0.105i)8-s + (0.559 + 0.828i)9-s + (−0.152 − 0.0605i)10-s + (−0.819 + 0.180i)11-s + (−0.763 − 0.518i)12-s + (−0.977 + 0.830i)13-s + (0.636 − 0.382i)14-s + (−0.0857 + 0.0811i)15-s + (−1.04 − 0.230i)16-s + (0.342 + 0.231i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.277045 - 1.66222i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.277045 - 1.66222i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.58 - 2.43i)T \) |
| 59 | \( 1 + (-76.9 - 446. i)T \) |
| good | 2 | \( 1 + (0.212 - 3.91i)T + (-7.95 - 0.864i)T^{2} \) |
| 5 | \( 1 + (0.421 - 1.25i)T + (-99.5 - 75.6i)T^{2} \) |
| 7 | \( 1 + (5.56 + 8.21i)T + (-126. + 318. i)T^{2} \) |
| 11 | \( 1 + (29.8 - 6.57i)T + (1.20e3 - 558. i)T^{2} \) |
| 13 | \( 1 + (45.8 - 38.9i)T + (355. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-23.9 - 16.2i)T + (1.81e3 + 4.56e3i)T^{2} \) |
| 19 | \( 1 + (4.96 - 9.36i)T + (-3.84e3 - 5.67e3i)T^{2} \) |
| 23 | \( 1 + (105. - 99.8i)T + (658. - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-169. + 9.19i)T + (2.42e4 - 2.63e3i)T^{2} \) |
| 31 | \( 1 + (-250. + 132. i)T + (1.67e4 - 2.46e4i)T^{2} \) |
| 37 | \( 1 + (-48.5 + 7.96i)T + (4.80e4 - 1.61e4i)T^{2} \) |
| 41 | \( 1 + (9.02 - 9.52i)T + (-3.73e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (-31.5 + 143. i)T + (-7.21e4 - 3.33e4i)T^{2} \) |
| 47 | \( 1 + (45.7 - 15.4i)T + (8.26e4 - 6.28e4i)T^{2} \) |
| 53 | \( 1 + (-461. + 183. i)T + (1.08e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-257. - 13.9i)T + (2.25e5 + 2.45e4i)T^{2} \) |
| 67 | \( 1 + (854. + 140. i)T + (2.85e5 + 9.60e4i)T^{2} \) |
| 71 | \( 1 + (-312. - 928. i)T + (-2.84e5 + 2.16e5i)T^{2} \) |
| 73 | \( 1 + (-35.8 - 59.5i)T + (-1.82e5 + 3.43e5i)T^{2} \) |
| 79 | \( 1 + (-779. - 360. i)T + (3.19e5 + 3.75e5i)T^{2} \) |
| 83 | \( 1 + (68.8 + 247. i)T + (-4.89e5 + 2.94e5i)T^{2} \) |
| 89 | \( 1 + (-52.8 - 975. i)T + (-7.00e5 + 7.62e4i)T^{2} \) |
| 97 | \( 1 + (-229. + 381. i)T + (-4.27e5 - 8.06e5i)T^{2} \) |
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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
−13.33147383882261258291708661299, −11.86418621202151902138542820188, −10.35136059311146627231024072785, −9.583600383878515054227645097283, −8.417124797526211085075298244876, −7.61219696414876936077500904746, −6.80709196595865360007466522443, −5.32857940215870540070418280825, −4.21710220434994493585127454208, −2.51837491495274261906986644380,
0.68475775247018526286005804498, 2.46767666972595338779680565139, 3.01331642564426741929901714278, 4.65023611166381009985756715330, 6.47757777241792570810202906138, 7.87316458281088861328283704639, 8.793569506726935119069614381037, 9.958653833201224852886482269670, 10.46650850661197246465380221959, 12.13845876582265159267214441848
