| L(s) = 1 | + (0.186 − 0.982i)2-s + (0.0181 + 0.000425i)3-s + (−0.930 − 0.366i)4-s + (−0.713 − 2.70i)5-s + (0.00380 − 0.0177i)6-s + (1.80 − 4.02i)7-s + (−0.533 + 0.845i)8-s + (−2.99 − 0.140i)9-s + (−2.79 + 0.196i)10-s + (−2.74 + 3.08i)11-s + (−0.0167 − 0.00704i)12-s + (0.108 + 1.54i)13-s + (−3.61 − 2.52i)14-s + (−0.0118 − 0.0494i)15-s + (0.731 + 0.681i)16-s + (−2.87 + 2.55i)17-s + ⋯ |
| L(s) = 1 | + (0.131 − 0.694i)2-s + (0.0104 + 0.000245i)3-s + (−0.465 − 0.183i)4-s + (−0.319 − 1.21i)5-s + (0.00155 − 0.00724i)6-s + (0.682 − 1.51i)7-s + (−0.188 + 0.299i)8-s + (−0.998 − 0.0468i)9-s + (−0.882 + 0.0622i)10-s + (−0.827 + 0.930i)11-s + (−0.00483 − 0.00203i)12-s + (0.0301 + 0.428i)13-s + (−0.965 − 0.674i)14-s + (−0.00304 − 0.0127i)15-s + (0.182 + 0.170i)16-s + (−0.697 + 0.620i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0425125 + 0.967594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0425125 + 0.967594i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.186 + 0.982i)T \) |
| 269 | \( 1 + (-16.3 - 1.18i)T \) |
| good | 3 | \( 1 + (-0.0181 - 0.000425i)T + (2.99 + 0.140i)T^{2} \) |
| 5 | \( 1 + (0.713 + 2.70i)T + (-4.34 + 2.46i)T^{2} \) |
| 7 | \( 1 + (-1.80 + 4.02i)T + (-4.65 - 5.23i)T^{2} \) |
| 11 | \( 1 + (2.74 - 3.08i)T + (-1.28 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.108 - 1.54i)T + (-12.8 + 1.82i)T^{2} \) |
| 17 | \( 1 + (2.87 - 2.55i)T + (1.98 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.07 + 2.70i)T + (7.37 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.0466 + 1.98i)T + (-22.9 - 1.07i)T^{2} \) |
| 29 | \( 1 + (-1.10 + 1.94i)T + (-14.8 - 24.8i)T^{2} \) |
| 31 | \( 1 + (1.43 - 0.169i)T + (30.1 - 7.20i)T^{2} \) |
| 37 | \( 1 + (0.326 + 0.608i)T + (-20.4 + 30.8i)T^{2} \) |
| 41 | \( 1 + (-0.468 + 0.0888i)T + (38.1 - 15.0i)T^{2} \) |
| 43 | \( 1 + (10.4 + 5.91i)T + (22.0 + 36.8i)T^{2} \) |
| 47 | \( 1 + (5.51 - 1.59i)T + (39.7 - 25.0i)T^{2} \) |
| 53 | \( 1 + (-5.74 + 7.83i)T + (-15.9 - 50.5i)T^{2} \) |
| 59 | \( 1 + (-0.566 + 1.43i)T + (-43.1 - 40.2i)T^{2} \) |
| 61 | \( 1 + (-5.45 + 6.74i)T + (-12.7 - 59.6i)T^{2} \) |
| 67 | \( 1 + (-6.90 + 2.71i)T + (49.0 - 45.6i)T^{2} \) |
| 71 | \( 1 + (-6.08 - 8.71i)T + (-24.4 + 66.6i)T^{2} \) |
| 73 | \( 1 + (4.77 + 13.0i)T + (-55.6 + 47.2i)T^{2} \) |
| 79 | \( 1 + (-11.9 + 9.20i)T + (20.1 - 76.3i)T^{2} \) |
| 83 | \( 1 + (1.39 - 14.8i)T + (-81.5 - 15.4i)T^{2} \) |
| 89 | \( 1 + (-2.40 + 14.5i)T + (-84.2 - 28.6i)T^{2} \) |
| 97 | \( 1 + (-3.27 - 4.04i)T + (-20.3 + 94.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.50791160211931857320254780073, −9.600689333272792828008747264843, −8.554198248201803942383676705017, −7.956033898110376881215044372098, −6.83706023575108433912086836680, −5.13036212060368556714482396726, −4.69071535920127714392281639716, −3.66271809894041066304055031367, −1.97975594316244933735286570183, −0.52757378545028359423182493846,
2.62054989245534308240101089336, 3.28449015692008147318777104657, 5.20468156872112387263659729982, 5.64145707883593347802543709203, 6.65853028171629013113739527261, 7.85061850593693538016664250999, 8.359668381767209544673956810016, 9.251883960891406824034743617696, 10.52918562469548305278143190260, 11.46203560939207826137621843106
