| L(s) = 1 | + (0.851 − 1.12i)2-s + (0.496 + 0.226i)3-s + (−0.549 − 1.92i)4-s + (1.16 + 1.00i)5-s + (0.679 − 0.367i)6-s + (0.981 − 0.630i)7-s + (−2.63 − 1.01i)8-s + (−1.76 − 2.04i)9-s + (2.13 − 0.455i)10-s + (4.28 + 0.616i)11-s + (0.163 − 1.08i)12-s + (−1.76 + 2.74i)13-s + (0.123 − 1.64i)14-s + (0.349 + 0.765i)15-s + (−3.39 + 2.11i)16-s + (−3.79 − 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.602 − 0.798i)2-s + (0.286 + 0.131i)3-s + (−0.274 − 0.961i)4-s + (0.520 + 0.451i)5-s + (0.277 − 0.150i)6-s + (0.370 − 0.238i)7-s + (−0.933 − 0.359i)8-s + (−0.589 − 0.680i)9-s + (0.674 − 0.144i)10-s + (1.29 + 0.185i)11-s + (0.0470 − 0.311i)12-s + (−0.489 + 0.761i)13-s + (0.0329 − 0.439i)14-s + (0.0903 + 0.197i)15-s + (−0.848 + 0.528i)16-s + (−0.919 − 0.270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47993 - 0.897537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47993 - 0.897537i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.851 + 1.12i)T \) |
| 23 | \( 1 + (1.22 - 4.63i)T \) |
| good | 3 | \( 1 + (-0.496 - 0.226i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-1.16 - 1.00i)T + (0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.981 + 0.630i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-4.28 - 0.616i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.76 - 2.74i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (3.79 + 1.11i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.469 - 1.59i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (1.96 - 6.68i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.28 - 2.81i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (2.91 - 2.52i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.80 + 5.54i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.605 + 0.276i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 + (5.72 + 8.90i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.863 + 1.34i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.81 - 0.829i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (2.22 - 0.320i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.69 + 11.7i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (6.66 - 1.95i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (4.03 + 2.59i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-13.6 + 11.8i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.74 + 16.9i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (0.978 - 1.12i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−12.21032780779657903159853268661, −11.62803606466942034107263327282, −10.61640867824575852116049055682, −9.476918813165541496629716325828, −8.931929163834108152510868781591, −6.94539904095809386383128091858, −6.01409416847253792936135063835, −4.53005198002899816415809009265, −3.38693258709444952119886030354, −1.86019897142078243660794568957,
2.47862708789176203838113145484, 4.23534526763655687379739246547, 5.39385663175529727131833729310, 6.34542685235857109157404934658, 7.68208110514105538545156120232, 8.607473778961649057068249587606, 9.380035759493717863679305342431, 11.06849431088878170019817596559, 12.05175506166525390246752279523, 13.08718671149588334881239078274
