| L(s) = 1 | + (−1.40 − 0.115i)2-s + (−0.121 + 0.105i)3-s + (1.97 + 0.324i)4-s + (2.18 − 0.314i)5-s + (0.183 − 0.134i)6-s + (−0.526 + 1.15i)7-s + (−2.74 − 0.685i)8-s + (−0.423 + 2.94i)9-s + (−3.11 + 0.190i)10-s + (1.19 − 4.06i)11-s + (−0.274 + 0.168i)12-s + (1.87 − 0.856i)13-s + (0.874 − 1.56i)14-s + (−0.232 + 0.268i)15-s + (3.78 + 1.28i)16-s + (6.16 + 3.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 − 0.0814i)2-s + (−0.0701 + 0.0608i)3-s + (0.986 + 0.162i)4-s + (0.976 − 0.140i)5-s + (0.0749 − 0.0549i)6-s + (−0.198 + 0.435i)7-s + (−0.970 − 0.242i)8-s + (−0.141 + 0.981i)9-s + (−0.985 + 0.0603i)10-s + (0.359 − 1.22i)11-s + (−0.0791 + 0.0486i)12-s + (0.520 − 0.237i)13-s + (0.233 − 0.417i)14-s + (−0.0600 + 0.0692i)15-s + (0.947 + 0.320i)16-s + (1.49 + 0.961i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.890126 + 0.0783999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.890126 + 0.0783999i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.115i)T \) |
| 23 | \( 1 + (3.19 + 3.57i)T \) |
| good | 3 | \( 1 + (0.121 - 0.105i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-2.18 + 0.314i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.526 - 1.15i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.19 + 4.06i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.87 + 0.856i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-6.16 - 3.96i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-2.67 - 4.15i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.63 + 5.65i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.04 + 1.20i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (7.32 + 1.05i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.490 + 3.41i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (8.83 - 7.65i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 5.20T + 47T^{2} \) |
| 53 | \( 1 + (5.98 + 2.73i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.35 + 1.53i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.450 - 0.390i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (2.97 + 10.1i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (5.73 - 1.68i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (3.25 - 2.09i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.14 + 2.51i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (5.66 + 0.813i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (2.88 + 3.33i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.493 + 3.43i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−12.44665406750240117821814299733, −11.47584482375740030059392950680, −10.35012632242377842444865685933, −9.849273739584673558412720178304, −8.511152444845743803149319864264, −7.975943752546272049294363658113, −6.13497576412023160458075837568, −5.70814516515386929076404863168, −3.25461680181575741478480047422, −1.65818617248217980789149431746,
1.44079435788058140398765459411, 3.23741960711960641347087691967, 5.40409579407833619236254297875, 6.64771150684000328802550305148, 7.27218985681361124879412587724, 8.822831022316767226810669584352, 9.789944051679349186317781865002, 10.08157499332976104426123927071, 11.62182355950239940443361521333, 12.24794606358040046115383455925
