L(s) = 1 | + (0.940 + 1.49i)2-s + (−0.294 + 0.955i)3-s + (−0.922 + 1.91i)4-s + (−1.70 + 0.457i)6-s + (−1.97 + 0.223i)8-s + (−0.826 − 0.563i)9-s + (−1.55 − 1.44i)12-s + (−1.49 + 1.19i)13-s + (−0.870 − 1.09i)16-s + (0.0661 − 1.76i)18-s + (0.974 − 0.222i)23-s + (0.370 − 1.95i)24-s + (−0.900 − 0.433i)25-s + (−3.18 − 1.11i)26-s + (0.781 − 0.623i)27-s + ⋯ |
L(s) = 1 | + (0.940 + 1.49i)2-s + (−0.294 + 0.955i)3-s + (−0.922 + 1.91i)4-s + (−1.70 + 0.457i)6-s + (−1.97 + 0.223i)8-s + (−0.826 − 0.563i)9-s + (−1.55 − 1.44i)12-s + (−1.49 + 1.19i)13-s + (−0.870 − 1.09i)16-s + (0.0661 − 1.76i)18-s + (0.974 − 0.222i)23-s + (0.370 − 1.95i)24-s + (−0.900 − 0.433i)25-s + (−3.18 − 1.11i)26-s + (0.781 − 0.623i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.274295332\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.274295332\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.294 - 0.955i)T \) |
| 23 | \( 1 + (-0.974 + 0.222i)T \) |
| 29 | \( 1 + (-0.997 + 0.0747i)T \) |
good | 2 | \( 1 + (-0.940 - 1.49i)T + (-0.433 + 0.900i)T^{2} \) |
| 5 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 7 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 13 | \( 1 + (1.49 - 1.19i)T + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 31 | \( 1 + (1.63 - 1.02i)T + (0.433 - 0.900i)T^{2} \) |
| 37 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 41 | \( 1 + (-1.13 - 1.13i)T + iT^{2} \) |
| 43 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 47 | \( 1 + (-0.146 + 1.29i)T + (-0.974 - 0.222i)T^{2} \) |
| 53 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 - 1.80iT - T^{2} \) |
| 61 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 67 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 71 | \( 1 + (-0.848 - 1.06i)T + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-1.66 - 1.04i)T + (0.433 + 0.900i)T^{2} \) |
| 79 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 83 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 97 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
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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
−9.607279155038357484789751008104, −8.964070658379146163613722135653, −8.194113789545654045290916818791, −7.12061764005104217614802553443, −6.75729660623488769572183637444, −5.71677452110442095613420402921, −5.09195192658708625878642189237, −4.44589020372508515528893450723, −3.77622980518830526980438694708, −2.60105092615514459905410929078,
0.68140438543061815252031011930, 2.00781232090479108012899740836, 2.68485149845397789082808845166, 3.57893458811757370920675778753, 4.81487952063188905024389235400, 5.37786230613224279681895826705, 6.10123158987148333867174875181, 7.34823030915175378877372113997, 7.84395489426776019215160200275, 9.200389751447975067400349029142