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
−11.55110552994287573987102828968, −10.73060571169282915315504396986, −9.447437242418745027684205055413, −8.952142003714048190381085536514, −7.906970506004845402888079222603, −6.91861782787402441507069574961, −6.31185998811018061836473568634, −4.73631435794373853931786346021, −3.13239015551153897722638505271, −1.70978661733719066593483416251,
1.33646268000893366508660692738, 3.36738781035266669941280591695, 3.71410513904426677448086847186, 5.55956446516524632606525563618, 6.73775173880629070675675892695, 8.326516033739714165779976272124, 8.790573645310409443775301412617, 9.463559317122785402922856969455, 10.57571944603147580381096532507, 11.27678060733998962152505948594