| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−2.44 + 1.57i)5-s + (0.654 − 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.414 − 2.88i)10-s + (−1.74 − 3.82i)11-s + (0.415 + 0.909i)12-s + (0.00630 + 0.0438i)13-s + (0.841 + 0.540i)14-s + (2.79 − 0.820i)15-s + (−0.142 + 0.989i)16-s + (3.34 − 3.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (−1.09 + 0.703i)5-s + (0.267 − 0.308i)6-s + (0.0537 − 0.374i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.131 − 0.911i)10-s + (−0.526 − 1.15i)11-s + (0.119 + 0.262i)12-s + (0.00174 + 0.0121i)13-s + (0.224 + 0.144i)14-s + (0.721 − 0.211i)15-s + (−0.0355 + 0.247i)16-s + (0.810 − 0.935i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.513527 + 0.463313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.513527 + 0.463313i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (1.92 - 4.39i)T \) |
| good | 5 | \( 1 + (2.44 - 1.57i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (1.74 + 3.82i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.00630 - 0.0438i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.34 + 3.85i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.57 - 2.96i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (6.05 - 6.98i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-8.79 + 2.58i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (3.30 + 2.12i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (0.422 - 0.271i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-9.00 - 2.64i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 3.05T + 47T^{2} \) |
| 53 | \( 1 + (1.15 - 8.06i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.33 - 9.25i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-4.50 + 1.32i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (3.72 - 8.14i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (0.0557 - 0.122i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-6.49 - 7.49i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.394 - 2.74i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-12.9 - 8.31i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-18.0 - 5.29i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (1.20 - 0.776i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.29733044573002265584553036148, −9.377509981583073032834342843783, −8.169269138361687333020796557516, −7.59916668324396511682005367224, −7.07880163015231254192803452342, −5.93335335758560494604716295539, −5.26942810257146283663444403314, −3.96354097035591690704035659657, −3.07993435109703079461585268744, −0.917484165332207364489738843075,
0.53801968706211587040470285385, 2.10481098419747259683222129617, 3.56276349111131776310226115527, 4.50915553834395789703142552407, 5.11673175769818422073692331704, 6.40202041801530150058939494791, 7.67953989879907892092459422995, 8.080828919254778860323717134990, 9.105093865010949341544555880481, 9.956841410877758404311722734381
