| L(s) = 1 | + (2.78 + 0.491i)2-s + (7.97 − 4.17i)3-s + (7.51 + 2.73i)4-s + (−23.4 − 27.9i)5-s + (24.2 − 7.72i)6-s + (58.7 − 21.3i)7-s + (19.5 + 11.3i)8-s + (46.0 − 66.6i)9-s + (−51.5 − 89.2i)10-s + (−114. + 136. i)11-s + (71.3 − 9.59i)12-s + (25.3 + 143. i)13-s + (174. − 30.7i)14-s + (−303. − 124. i)15-s + (49.0 + 41.1i)16-s + (363. − 210. i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (0.885 − 0.464i)3-s + (0.469 + 0.171i)4-s + (−0.937 − 1.11i)5-s + (0.673 − 0.214i)6-s + (1.19 − 0.436i)7-s + (0.306 + 0.176i)8-s + (0.569 − 0.822i)9-s + (−0.515 − 0.892i)10-s + (−0.943 + 1.12i)11-s + (0.495 − 0.0666i)12-s + (0.150 + 0.851i)13-s + (0.888 − 0.156i)14-s + (−1.34 − 0.554i)15-s + (0.191 + 0.160i)16-s + (1.25 − 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.61412 - 0.960835i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.61412 - 0.960835i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.78 - 0.491i)T \) |
| 3 | \( 1 + (-7.97 + 4.17i)T \) |
| good | 5 | \( 1 + (23.4 + 27.9i)T + (-108. + 615. i)T^{2} \) |
| 7 | \( 1 + (-58.7 + 21.3i)T + (1.83e3 - 1.54e3i)T^{2} \) |
| 11 | \( 1 + (114. - 136. i)T + (-2.54e3 - 1.44e4i)T^{2} \) |
| 13 | \( 1 + (-25.3 - 143. i)T + (-2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-363. + 210. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (40.2 - 69.7i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (323. - 887. i)T + (-2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (1.28e3 + 225. i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (115. + 41.9i)T + (7.07e5 + 5.93e5i)T^{2} \) |
| 37 | \( 1 + (-328. - 568. i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-26.0 + 4.59i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-1.34e3 - 1.12e3i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-275. - 755. i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + 2.77e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (2.10e3 + 2.51e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-55.3 + 20.1i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-295. - 1.67e3i)T + (-1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-2.85e3 + 1.64e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-1.62e3 + 2.81e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.37e3 + 7.82e3i)T + (-3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-6.09e3 - 1.07e3i)T + (4.45e7 + 1.62e7i)T^{2} \) |
| 89 | \( 1 + (4.02e3 + 2.32e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-2.46e3 - 2.06e3i)T + (1.53e7 + 8.71e7i)T^{2} \) |
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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
−14.42395107284523878533411675730, −13.38294061889321814660471183469, −12.34976483527744735054493987382, −11.51347293311695537317820899290, −9.502604627598736023002055599309, −7.81130814294072674754870461612, −7.62600500028976010609840248806, −5.04740455517239583649715712628, −3.92832203313548082227733487514, −1.63923940999439889704991049116,
2.66176439347238786791692276695, 3.82515101263995699412650808240, 5.52398850361958431105932315557, 7.66439728714805377745307901867, 8.308985152217332530385628658324, 10.57056264837919883319657594528, 11.00902034573447595281654554851, 12.52057428421867149715031471589, 13.97448373046812893258132942400, 14.79508324659341492123027557103
