| L(s) = 1 | + (−0.281 + 5.19i)2-s + (−5.12 + 0.876i)3-s + (−19.0 − 2.06i)4-s + (2.67 − 7.95i)5-s + (−3.11 − 26.8i)6-s + (1.88 + 2.77i)7-s + (9.36 − 57.1i)8-s + (25.4 − 8.97i)9-s + (40.5 + 16.1i)10-s + (−16.4 + 3.61i)11-s + (99.1 − 6.06i)12-s + (17.9 − 15.2i)13-s + (−14.9 + 9.00i)14-s + (−6.75 + 43.0i)15-s + (144. + 31.8i)16-s + (−10.8 − 7.33i)17-s + ⋯ |
| L(s) = 1 | + (−0.0996 + 1.83i)2-s + (−0.985 + 0.168i)3-s + (−2.37 − 0.258i)4-s + (0.239 − 0.711i)5-s + (−0.211 − 1.82i)6-s + (0.101 + 0.149i)7-s + (0.413 − 2.52i)8-s + (0.943 − 0.332i)9-s + (1.28 + 0.511i)10-s + (−0.449 + 0.0990i)11-s + (2.38 − 0.146i)12-s + (0.383 − 0.325i)13-s + (−0.285 + 0.171i)14-s + (−0.116 + 0.741i)15-s + (2.26 + 0.498i)16-s + (−0.154 − 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.734463 + 0.514893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.734463 + 0.514893i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.12 - 0.876i)T \) |
| 59 | \( 1 + (-178. - 416. i)T \) |
| good | 2 | \( 1 + (0.281 - 5.19i)T + (-7.95 - 0.864i)T^{2} \) |
| 5 | \( 1 + (-2.67 + 7.95i)T + (-99.5 - 75.6i)T^{2} \) |
| 7 | \( 1 + (-1.88 - 2.77i)T + (-126. + 318. i)T^{2} \) |
| 11 | \( 1 + (16.4 - 3.61i)T + (1.20e3 - 558. i)T^{2} \) |
| 13 | \( 1 + (-17.9 + 15.2i)T + (355. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (10.8 + 7.33i)T + (1.81e3 + 4.56e3i)T^{2} \) |
| 19 | \( 1 + (-13.4 + 25.3i)T + (-3.84e3 - 5.67e3i)T^{2} \) |
| 23 | \( 1 + (-15.5 + 14.6i)T + (658. - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (137. - 7.44i)T + (2.42e4 - 2.63e3i)T^{2} \) |
| 31 | \( 1 + (-291. + 154. i)T + (1.67e4 - 2.46e4i)T^{2} \) |
| 37 | \( 1 + (-124. + 20.4i)T + (4.80e4 - 1.61e4i)T^{2} \) |
| 41 | \( 1 + (-292. + 308. i)T + (-3.73e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (68.4 - 311. i)T + (-7.21e4 - 3.33e4i)T^{2} \) |
| 47 | \( 1 + (415. - 139. i)T + (8.26e4 - 6.28e4i)T^{2} \) |
| 53 | \( 1 + (-341. + 135. i)T + (1.08e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-685. - 37.1i)T + (2.25e5 + 2.45e4i)T^{2} \) |
| 67 | \( 1 + (-862. - 141. i)T + (2.85e5 + 9.60e4i)T^{2} \) |
| 71 | \( 1 + (135. + 402. i)T + (-2.84e5 + 2.16e5i)T^{2} \) |
| 73 | \( 1 + (308. + 513. i)T + (-1.82e5 + 3.43e5i)T^{2} \) |
| 79 | \( 1 + (231. + 107. i)T + (3.19e5 + 3.75e5i)T^{2} \) |
| 83 | \( 1 + (185. + 666. i)T + (-4.89e5 + 2.94e5i)T^{2} \) |
| 89 | \( 1 + (55.8 + 1.03e3i)T + (-7.00e5 + 7.62e4i)T^{2} \) |
| 97 | \( 1 + (-162. + 270. i)T + (-4.27e5 - 8.06e5i)T^{2} \) |
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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
−12.91217892089724386908506063459, −11.47311523656661591554623230999, −10.07466756497087401058750856805, −9.159992040164351626987136470143, −8.122687270005381582696853700278, −7.05243676828167907535674790010, −5.97848560082846554114588952936, −5.25065430122736873798715378579, −4.34773798146815821667990218969, −0.63475818127410599641780323657,
1.06109927944061338935196375920, 2.51415286453337394580281895393, 3.97751925677551728125255271735, 5.20495265866580203587850545353, 6.63681787528458136981663488155, 8.207783958432909162688604340942, 9.659817653930390618069587770151, 10.39235048789418348298363983959, 11.09861189207713633245240946684, 11.74668843819689694065743230961
