| L(s) = 1 | + (−3.82 − 1.16i)2-s + 9.98i·3-s + (13.2 + 8.91i)4-s + 35.6·5-s + (11.6 − 38.2i)6-s − 14.7i·7-s + (−40.4 − 49.6i)8-s − 18.7·9-s + (−136. − 41.5i)10-s + 42.5i·11-s + (−89.0 + 132. i)12-s + 173.·13-s + (−17.2 + 56.5i)14-s + 355. i·15-s + (96.9 + 236. i)16-s + 90.6·17-s + ⋯ |
| L(s) = 1 | + (−0.956 − 0.291i)2-s + 1.10i·3-s + (0.830 + 0.557i)4-s + 1.42·5-s + (0.323 − 1.06i)6-s − 0.301i·7-s + (−0.631 − 0.775i)8-s − 0.231·9-s + (−1.36 − 0.415i)10-s + 0.351i·11-s + (−0.618 + 0.921i)12-s + 1.02·13-s + (−0.0878 + 0.288i)14-s + 1.58i·15-s + (0.378 + 0.925i)16-s + 0.313·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.25315 + 0.668123i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25315 + 0.668123i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.82 + 1.16i)T \) |
| 19 | \( 1 + 82.8iT \) |
| good | 3 | \( 1 - 9.98iT - 81T^{2} \) |
| 5 | \( 1 - 35.6T + 625T^{2} \) |
| 7 | \( 1 + 14.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 42.5iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 173.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 90.6T + 8.35e4T^{2} \) |
| 23 | \( 1 - 542. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 624.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.35e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 716.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 268.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.11e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 3.63e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.64e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.78e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 7.08e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.30e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 1.95e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 494.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.05e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.71e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.11e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 8.56e3T + 8.85e7T^{2} \) |
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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
−13.94616916733522401951349610490, −12.81343544432168892759985802687, −11.19160186545509221875321484269, −10.26851831452986693311813121023, −9.657042584915688575837575225762, −8.721635514953998791519712251762, −6.94798540582599444004496063581, −5.48102681696296370929145544848, −3.55648269515268295051698184240, −1.63905508308086795866472020218,
1.14859047092395146361811970087, 2.31336412242529499784131073114, 5.82690783149825150206650942456, 6.42618195721123740255412505664, 7.81417066921920616396644966570, 8.965152219356338940368560741057, 10.03132692017893272792628082781, 11.22288604941945083127807482492, 12.62500733352037232065907403339, 13.58169319158528519169940320209
