| L(s) = 1 | + (−0.654 − 0.755i)2-s + (1.33 + 0.855i)3-s + (−0.142 + 0.989i)4-s + (0.426 − 0.934i)5-s + (−0.225 − 1.56i)6-s + (−0.959 − 0.281i)7-s + (0.841 − 0.540i)8-s + (−0.205 − 0.449i)9-s + (−0.986 + 0.289i)10-s + (2.66 − 3.07i)11-s + (−1.03 + 1.19i)12-s + (5.08 − 1.49i)13-s + (0.415 + 0.909i)14-s + (1.36 − 0.879i)15-s + (−0.959 − 0.281i)16-s + (0.323 + 2.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (0.768 + 0.494i)3-s + (−0.0711 + 0.494i)4-s + (0.190 − 0.418i)5-s + (−0.0919 − 0.639i)6-s + (−0.362 − 0.106i)7-s + (0.297 − 0.191i)8-s + (−0.0683 − 0.149i)9-s + (−0.311 + 0.0915i)10-s + (0.804 − 0.928i)11-s + (−0.299 + 0.345i)12-s + (1.40 − 0.413i)13-s + (0.111 + 0.243i)14-s + (0.353 − 0.227i)15-s + (−0.239 − 0.0704i)16-s + (0.0785 + 0.546i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31304 - 0.450791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31304 - 0.450791i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (-4.65 - 1.16i)T \) |
| good | 3 | \( 1 + (-1.33 - 0.855i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (-0.426 + 0.934i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (-2.66 + 3.07i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-5.08 + 1.49i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.323 - 2.25i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.474 - 3.29i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.0814 - 0.566i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (8.38 - 5.38i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (0.813 + 1.78i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.46 + 5.40i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (9.19 + 5.91i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 7.21T + 47T^{2} \) |
| 53 | \( 1 + (7.15 + 2.10i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-4.02 + 1.18i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (5.62 - 3.61i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (2.06 + 2.38i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-6.56 - 7.58i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.30 - 9.09i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (11.2 - 3.29i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-6.21 - 13.6i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-11.6 - 7.50i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.04 + 6.67i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.27678060733998962152505948594, −10.57571944603147580381096532507, −9.463559317122785402922856969455, −8.790573645310409443775301412617, −8.326516033739714165779976272124, −6.73775173880629070675675892695, −5.55956446516524632606525563618, −3.71410513904426677448086847186, −3.36738781035266669941280591695, −1.33646268000893366508660692738,
1.70978661733719066593483416251, 3.13239015551153897722638505271, 4.73631435794373853931786346021, 6.31185998811018061836473568634, 6.91861782787402441507069574961, 7.906970506004845402888079222603, 8.952142003714048190381085536514, 9.447437242418745027684205055413, 10.73060571169282915315504396986, 11.55110552994287573987102828968
