| L(s) = 1 | + (−1.22 + 0.700i)2-s + (−1.98 + 1.66i)3-s + (1.01 − 1.72i)4-s + (−2.53 − 0.920i)5-s + (1.27 − 3.43i)6-s + (−1.43 − 0.829i)7-s + (−0.0475 + 2.82i)8-s + (0.645 − 3.66i)9-s + (3.75 − 0.640i)10-s + (−2.94 + 1.69i)11-s + (0.842 + 5.11i)12-s + (−2.17 + 2.59i)13-s + (2.34 + 0.0131i)14-s + (6.55 − 2.38i)15-s + (−1.92 − 3.50i)16-s + (1.03 + 5.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.868 + 0.495i)2-s + (−1.14 + 0.961i)3-s + (0.509 − 0.860i)4-s + (−1.13 − 0.411i)5-s + (0.519 − 1.40i)6-s + (−0.543 − 0.313i)7-s + (−0.0168 + 0.999i)8-s + (0.215 − 1.22i)9-s + (1.18 − 0.202i)10-s + (−0.886 + 0.511i)11-s + (0.243 + 1.47i)12-s + (−0.603 + 0.719i)13-s + (0.627 + 0.00351i)14-s + (1.69 − 0.616i)15-s + (−0.480 − 0.877i)16-s + (0.251 + 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0215577 - 0.119351i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0215577 - 0.119351i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 - 0.700i)T \) |
| 19 | \( 1 + (-4.28 - 0.772i)T \) |
| good | 3 | \( 1 + (1.98 - 1.66i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (2.53 + 0.920i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.43 + 0.829i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.94 - 1.69i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.17 - 2.59i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.03 - 5.88i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (2.01 + 5.52i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (2.92 + 0.516i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.91 - 5.04i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.798iT - 37T^{2} \) |
| 41 | \( 1 + (-0.904 - 1.07i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.235 - 0.645i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.47 - 1.14i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (2.91 + 7.99i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.838 - 4.75i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.56 + 1.29i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.76 - 10.0i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.34 - 0.855i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (8.03 - 6.73i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (6.98 - 5.86i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (6.78 + 3.91i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.99 - 9.52i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (8.40 - 1.48i)T + (91.1 - 33.1i)T^{2} \) |
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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
−15.64179305958085924362821155393, −14.62788822206527579655649381668, −12.53035960462218398259685833321, −11.55680059322227078004534946202, −10.49342321913859501060933413380, −9.787267205725826763100694040302, −8.272972375864558513665889705554, −7.02082686354741365429927318283, −5.53688392553503285038041023974, −4.27559927356433432435737889938,
0.22787933271959085643020465799, 3.07073656979986869316012514807, 5.64256002024942142324789354652, 7.35460989617547512800071361976, 7.63892250698200593876822173903, 9.554630516502136387183435969857, 10.95936922330593138306551824357, 11.65350850058653750020576640509, 12.35652765455510766795535701042, 13.39654564794834624169974204832
