| L(s) = 1 | + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−1.36 + 2.98i)5-s + (0.142 + 0.989i)6-s + (0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (3.14 − 0.923i)10-s + (3.20 − 3.69i)11-s + (0.654 − 0.755i)12-s + (−2.28 + 0.669i)13-s + (−0.415 − 0.909i)14-s + (2.75 − 1.77i)15-s + (−0.959 − 0.281i)16-s + (0.570 + 3.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (−0.608 + 1.33i)5-s + (0.0580 + 0.404i)6-s + (0.362 + 0.106i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (0.994 − 0.292i)10-s + (0.965 − 1.11i)11-s + (0.189 − 0.218i)12-s + (−0.632 + 0.185i)13-s + (−0.111 − 0.243i)14-s + (0.711 − 0.457i)15-s + (−0.239 − 0.0704i)16-s + (0.138 + 0.962i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.512405 + 0.439612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.512405 + 0.439612i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (1.45 - 4.57i)T \) |
| good | 5 | \( 1 + (1.36 - 2.98i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (-3.20 + 3.69i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.28 - 0.669i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.570 - 3.96i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.889 + 6.18i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.755 - 5.25i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (0.451 - 0.290i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.42 - 7.50i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (4.22 - 9.25i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (3.22 + 2.06i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + (-8.83 - 2.59i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (5.36 - 1.57i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (11.5 - 7.39i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (4.82 + 5.57i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-4.57 - 5.27i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.108 + 0.755i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-13.8 + 4.06i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-6.66 - 14.6i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (7.66 + 4.92i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.95 - 6.47i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.45745332073664895252882484537, −9.485268590717529420851200269680, −8.503787604908738137631479702985, −7.69285211759470831674548140375, −6.84855281102598246569706740218, −6.24001977849942878624064723206, −4.85768232212499709577857925766, −3.61760621624628526303881807770, −2.85915102804316376078341201029, −1.37988138023371888532746351105,
0.42577963210828414682423895826, 1.77855587215808086703260464050, 3.96903584908025776958477391734, 4.66527498632165098309607500479, 5.34404710481829345259888650921, 6.46342811814141648815597837915, 7.49118391313660597592517159828, 8.082500524977117253137091389090, 9.073476435609344176611571158968, 9.656834486675196511479130583871
