| L(s) = 1 | + (−0.329 + 1.37i)2-s + (0.583 − 0.908i)3-s + (−1.78 − 0.907i)4-s + (0.293 + 0.133i)5-s + (1.05 + 1.10i)6-s + (1.64 + 0.483i)7-s + (1.83 − 2.15i)8-s + (0.761 + 1.66i)9-s + (−0.280 + 0.359i)10-s + (4.72 + 4.09i)11-s + (−1.86 + 1.08i)12-s + (−0.960 − 3.26i)13-s + (−1.20 + 2.10i)14-s + (0.292 − 0.188i)15-s + (2.35 + 3.23i)16-s + (−0.348 − 2.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.233 + 0.972i)2-s + (0.337 − 0.524i)3-s + (−0.891 − 0.453i)4-s + (0.131 + 0.0598i)5-s + (0.431 + 0.450i)6-s + (0.622 + 0.182i)7-s + (0.649 − 0.760i)8-s + (0.253 + 0.555i)9-s + (−0.0888 + 0.113i)10-s + (1.42 + 1.23i)11-s + (−0.538 + 0.314i)12-s + (−0.266 − 0.906i)13-s + (−0.323 + 0.562i)14-s + (0.0756 − 0.0486i)15-s + (0.588 + 0.808i)16-s + (−0.0845 − 0.588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08295 + 0.505684i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08295 + 0.505684i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.329 - 1.37i)T \) |
| 23 | \( 1 + (4.35 + 2.00i)T \) |
| good | 3 | \( 1 + (-0.583 + 0.908i)T + (-1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (-0.293 - 0.133i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-1.64 - 0.483i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-4.72 - 4.09i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.960 + 3.26i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.348 + 2.42i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.170 - 0.0244i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (1.83 - 0.264i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (7.35 - 4.72i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (2.57 - 1.17i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.22 + 2.69i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.434 + 0.675i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 + (-1.61 + 5.50i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (0.382 + 1.30i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (6.21 + 9.67i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (9.79 - 8.48i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-3.39 - 3.91i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.89 - 13.1i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (6.04 - 1.77i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-4.83 + 2.20i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-7.15 - 4.59i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.69 + 8.08i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−12.88669632687457698140932225694, −12.05064470464309051785525639626, −10.49802695510655906128757894687, −9.547037447635252209829531236766, −8.480152220119629566107207412250, −7.53608242722253776645992021293, −6.80963425409640152463528701997, −5.39817457969638900774446718888, −4.25317447578587949124532023553, −1.82107760469316949404228705575,
1.59469240333177229864014891638, 3.60367856814318891589349861612, 4.25383723483705585735847313274, 5.98868793636329392334392328600, 7.66559286686937080153087069163, 9.095745807549389002835616831959, 9.216919159850950608081711716130, 10.58204915259417387640258867888, 11.49587838577162841425298048373, 12.13666272132277896809030652129
