| L(s) = 1 | + (0.120 − 0.232i)2-s + (1.12 + 1.57i)4-s + (−0.991 + 4.08i)5-s + (3.50 + 1.21i)7-s + (1.01 − 0.146i)8-s + (0.832 + 0.721i)10-s + (0.185 − 3.88i)11-s + (−0.0862 − 0.249i)13-s + (0.703 − 0.670i)14-s + (−1.17 + 3.39i)16-s + (−1.22 − 2.67i)17-s + (1.48 + 0.679i)19-s + (−7.54 + 3.01i)20-s + (−0.882 − 0.509i)22-s + (−0.324 + 4.78i)23-s + ⋯ |
| L(s) = 1 | + (0.0848 − 0.164i)2-s + (0.560 + 0.786i)4-s + (−0.443 + 1.82i)5-s + (1.32 + 0.458i)7-s + (0.360 − 0.0518i)8-s + (0.263 + 0.228i)10-s + (0.0558 − 1.17i)11-s + (−0.0239 − 0.0691i)13-s + (0.188 − 0.179i)14-s + (−0.293 + 0.848i)16-s + (−0.295 − 0.647i)17-s + (0.341 + 0.155i)19-s + (−1.68 + 0.675i)20-s + (−0.188 − 0.108i)22-s + (−0.0677 + 0.997i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0519 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0519 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33277 + 1.26528i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33277 + 1.26528i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (0.324 - 4.78i)T \) |
| good | 2 | \( 1 + (-0.120 + 0.232i)T + (-1.16 - 1.62i)T^{2} \) |
| 5 | \( 1 + (0.991 - 4.08i)T + (-4.44 - 2.29i)T^{2} \) |
| 7 | \( 1 + (-3.50 - 1.21i)T + (5.50 + 4.32i)T^{2} \) |
| 11 | \( 1 + (-0.185 + 3.88i)T + (-10.9 - 1.04i)T^{2} \) |
| 13 | \( 1 + (0.0862 + 0.249i)T + (-10.2 + 8.03i)T^{2} \) |
| 17 | \( 1 + (1.22 + 2.67i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.48 - 0.679i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.00 - 1.42i)T + (9.48 + 27.4i)T^{2} \) |
| 31 | \( 1 + (0.710 + 0.284i)T + (22.4 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-1.05 + 3.58i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (3.88 + 0.941i)T + (36.4 + 18.7i)T^{2} \) |
| 43 | \( 1 + (2.64 + 6.61i)T + (-31.1 + 29.6i)T^{2} \) |
| 47 | \( 1 + (-4.75 + 2.74i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.42 - 1.64i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-9.13 + 3.16i)T + (46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (-2.52 - 3.21i)T + (-14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (13.3 - 0.637i)T + (66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (0.327 + 0.510i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.89 + 15.0i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.400 - 2.07i)T + (-73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-2.85 - 11.7i)T + (-73.7 + 38.0i)T^{2} \) |
| 89 | \( 1 + (-1.15 + 8.01i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.17 + 4.37i)T + (-4.61 - 96.8i)T^{2} \) |
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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
−10.97502652968422601204291254120, −10.42844747237807031861024354312, −8.881509978351563401713528598428, −7.945453729948415605061293718997, −7.40138303631258002653732044638, −6.54066209411749862788035788211, −5.42580404217630328636780920031, −3.86950052396738761928531946703, −3.08611368497009819503618609202, −2.12971770369512310452812108466,
1.07686450859310849023422401995, 1.94241545313731940222602802417, 4.42827386619377180766859870218, 4.65606601842218299302952779953, 5.60267802507206813054504656249, 6.91824938748180603434296048362, 7.88025940870943924914582194220, 8.501397619494911210435891593069, 9.546568202922078597929754420570, 10.39310082479507507702807700820
