| L(s) = 1 | + (−0.181 + 3.33i)2-s + (−4.07 − 3.22i)3-s + (−3.16 − 0.344i)4-s + (5.90 − 17.5i)5-s + (11.5 − 13.0i)6-s + (−10.9 − 16.1i)7-s + (−2.60 + 15.8i)8-s + (6.20 + 26.2i)9-s + (57.4 + 22.8i)10-s + (−44.9 + 9.89i)11-s + (11.7 + 11.6i)12-s + (−47.2 + 40.1i)13-s + (55.8 − 33.6i)14-s + (−80.5 + 52.3i)15-s + (−77.4 − 17.0i)16-s + (92.6 + 62.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.0640 + 1.18i)2-s + (−0.784 − 0.620i)3-s + (−0.395 − 0.0430i)4-s + (0.527 − 1.56i)5-s + (0.782 − 0.886i)6-s + (−0.590 − 0.871i)7-s + (−0.115 + 0.702i)8-s + (0.229 + 0.973i)9-s + (1.81 + 0.723i)10-s + (−1.23 + 0.271i)11-s + (0.283 + 0.279i)12-s + (−1.00 + 0.855i)13-s + (1.06 − 0.641i)14-s + (−1.38 + 0.900i)15-s + (−1.21 − 0.266i)16-s + (1.32 + 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0121634 - 0.146865i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0121634 - 0.146865i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.07 + 3.22i)T \) |
| 59 | \( 1 + (84.1 + 445. i)T \) |
| good | 2 | \( 1 + (0.181 - 3.33i)T + (-7.95 - 0.864i)T^{2} \) |
| 5 | \( 1 + (-5.90 + 17.5i)T + (-99.5 - 75.6i)T^{2} \) |
| 7 | \( 1 + (10.9 + 16.1i)T + (-126. + 318. i)T^{2} \) |
| 11 | \( 1 + (44.9 - 9.89i)T + (1.20e3 - 558. i)T^{2} \) |
| 13 | \( 1 + (47.2 - 40.1i)T + (355. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-92.6 - 62.7i)T + (1.81e3 + 4.56e3i)T^{2} \) |
| 19 | \( 1 + (59.2 - 111. i)T + (-3.84e3 - 5.67e3i)T^{2} \) |
| 23 | \( 1 + (-26.4 + 25.0i)T + (658. - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-47.2 + 2.56i)T + (2.42e4 - 2.63e3i)T^{2} \) |
| 31 | \( 1 + (86.0 - 45.6i)T + (1.67e4 - 2.46e4i)T^{2} \) |
| 37 | \( 1 + (8.67 - 1.42i)T + (4.80e4 - 1.61e4i)T^{2} \) |
| 41 | \( 1 + (-159. + 168. i)T + (-3.73e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (-1.92 + 8.72i)T + (-7.21e4 - 3.33e4i)T^{2} \) |
| 47 | \( 1 + (516. - 173. i)T + (8.26e4 - 6.28e4i)T^{2} \) |
| 53 | \( 1 + (349. - 139. i)T + (1.08e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (463. + 25.1i)T + (2.25e5 + 2.45e4i)T^{2} \) |
| 67 | \( 1 + (171. + 28.1i)T + (2.85e5 + 9.60e4i)T^{2} \) |
| 71 | \( 1 + (265. + 787. i)T + (-2.84e5 + 2.16e5i)T^{2} \) |
| 73 | \( 1 + (-140. - 234. i)T + (-1.82e5 + 3.43e5i)T^{2} \) |
| 79 | \( 1 + (-188. - 87.3i)T + (3.19e5 + 3.75e5i)T^{2} \) |
| 83 | \( 1 + (-196. - 707. i)T + (-4.89e5 + 2.94e5i)T^{2} \) |
| 89 | \( 1 + (56.5 + 1.04e3i)T + (-7.00e5 + 7.62e4i)T^{2} \) |
| 97 | \( 1 + (780. - 1.29e3i)T + (-4.27e5 - 8.06e5i)T^{2} \) |
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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
−12.63825959747790855797744438800, −12.30781980925514964472059079222, −10.62293101082755794214106531557, −9.676010155347780971561794511283, −8.192089613373446362300194039581, −7.58342697251570068614242260948, −6.38715314868851864517036151433, −5.49049634444819223459779317962, −4.64902818491456836303816459161, −1.73743291585498502273072009353,
0.06951701481028646545401149999, 2.73808799249902283133061903374, 3.03137688763035121785342348884, 5.16539989797135381676324171108, 6.22220593284010277524824104596, 7.31187472137167808301350159711, 9.455986849589037156960846070924, 10.04011904570297112258101859466, 10.71483446012893609518358525240, 11.43988714099563393912172817070
