| L(s) = 1 | + (0.592 − 1.28i)2-s + (−0.468 + 3.26i)3-s + (−1.29 − 1.52i)4-s + (−2.28 + 0.670i)5-s + (3.90 + 2.53i)6-s + (2.41 + 2.79i)7-s + (−2.72 + 0.765i)8-s + (−7.53 − 2.21i)9-s + (−0.491 + 3.33i)10-s + (−0.743 + 1.15i)11-s + (5.56 − 3.51i)12-s + (2.83 + 2.46i)13-s + (5.01 − 1.45i)14-s + (−1.11 − 7.76i)15-s + (−0.629 + 3.95i)16-s + (3.07 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (0.418 − 0.908i)2-s + (−0.270 + 1.88i)3-s + (−0.649 − 0.760i)4-s + (−1.02 + 0.299i)5-s + (1.59 + 1.03i)6-s + (0.914 + 1.05i)7-s + (−0.962 + 0.270i)8-s + (−2.51 − 0.737i)9-s + (−0.155 + 1.05i)10-s + (−0.224 + 0.349i)11-s + (1.60 − 1.01i)12-s + (0.787 + 0.682i)13-s + (1.34 − 0.388i)14-s + (−0.288 − 2.00i)15-s + (−0.157 + 0.987i)16-s + (0.746 − 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.782883 + 0.644977i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.782883 + 0.644977i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.592 + 1.28i)T \) |
| 23 | \( 1 + (-1.17 + 4.65i)T \) |
| good | 3 | \( 1 + (0.468 - 3.26i)T + (-2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (2.28 - 0.670i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-2.41 - 2.79i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (0.743 - 1.15i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.83 - 2.46i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.07 + 1.40i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.864 - 0.394i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.11 + 0.508i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-4.52 + 0.650i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (1.98 + 0.584i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (0.179 - 0.0527i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-5.68 - 0.816i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 6.84iT - 47T^{2} \) |
| 53 | \( 1 + (1.99 + 2.30i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (5.46 - 6.30i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.680 + 4.73i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.98 - 4.64i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-4.40 - 6.84i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.14 - 11.2i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.151 + 0.174i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-4.06 + 13.8i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-3.75 - 0.540i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (4.62 + 15.7i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−12.26813807712742863280329301203, −11.54421939295338314246713173250, −11.09275630378381623702501094727, −10.11399747045083942862439835435, −9.096130031207811939649662319760, −8.300546863747170693099662855352, −5.88291254324594856483539627248, −4.84392833646194794500656341758, −4.09766254793172716969075857607, −2.90003677314711078992528294485,
0.907016684843145767326118500064, 3.51289774899366005090739224232, 5.13615266691491830594661519136, 6.27075794482142397618124630865, 7.50512593507955906440739292285, 7.81617875611330471703019865214, 8.524943018250262745839207026428, 10.91184708733944703210836955368, 11.82081692352832524323860827490, 12.52800031871043541569278961201
