L(s) = 1 | + (0.642 − 0.766i)2-s + (−0.893 + 0.448i)3-s + (−0.173 − 0.984i)4-s + (−0.918 − 0.396i)5-s + (−0.230 + 0.973i)6-s + (−0.993 + 0.116i)7-s + (−0.866 − 0.5i)8-s + (0.597 − 0.802i)9-s + (−0.893 + 0.448i)10-s + (0.893 + 0.448i)11-s + (0.597 + 0.802i)12-s + (−0.116 − 0.993i)13-s + (−0.549 + 0.835i)14-s + (0.998 − 0.0581i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)2-s + (−0.893 + 0.448i)3-s + (−0.173 − 0.984i)4-s + (−0.918 − 0.396i)5-s + (−0.230 + 0.973i)6-s + (−0.993 + 0.116i)7-s + (−0.866 − 0.5i)8-s + (0.597 − 0.802i)9-s + (−0.893 + 0.448i)10-s + (0.893 + 0.448i)11-s + (0.597 + 0.802i)12-s + (−0.116 − 0.993i)13-s + (−0.549 + 0.835i)14-s + (0.998 − 0.0581i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09473301757 + 0.01029616638i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09473301757 + 0.01029616638i\) |
\(L(1)\) |
\(\approx\) |
\(0.5854929715 - 0.3989467323i\) |
\(L(1)\) |
\(\approx\) |
\(0.5854929715 - 0.3989467323i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 3 | \( 1 + (-0.893 + 0.448i)T \) |
| 5 | \( 1 + (-0.918 - 0.396i)T \) |
| 7 | \( 1 + (-0.993 + 0.116i)T \) |
| 11 | \( 1 + (0.893 + 0.448i)T \) |
| 13 | \( 1 + (-0.116 - 0.993i)T \) |
| 17 | \( 1 + (-0.342 - 0.939i)T \) |
| 19 | \( 1 + (0.984 + 0.173i)T \) |
| 23 | \( 1 + (-0.984 + 0.173i)T \) |
| 29 | \( 1 + (-0.998 - 0.0581i)T \) |
| 31 | \( 1 + (0.116 - 0.993i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.642 - 0.766i)T \) |
| 47 | \( 1 + (0.686 + 0.727i)T \) |
| 53 | \( 1 + (-0.396 + 0.918i)T \) |
| 59 | \( 1 + (0.998 - 0.0581i)T \) |
| 61 | \( 1 + (-0.727 - 0.686i)T \) |
| 67 | \( 1 + (-0.993 - 0.116i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 + (-0.893 - 0.448i)T \) |
| 79 | \( 1 + (0.802 + 0.597i)T \) |
| 83 | \( 1 + (-0.835 - 0.549i)T \) |
| 89 | \( 1 + (-0.230 + 0.973i)T \) |
| 97 | \( 1 + (0.802 - 0.597i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.22490733284308599595835649188, −17.392811624589842128279939043254, −16.67409409440792605026363387714, −16.210727394907669096325725587062, −15.835965591693025070012551257365, −14.872023969521686438954813942673, −14.200331278666999991316389303716, −13.48720873487805884001482534574, −12.80901165138249344681634848287, −12.043129932182150144882883654728, −11.71525070668704277074305832426, −11.019447456384295296431086828, −10.030472893252139785043019724, −9.08175241039954327966601914392, −8.29905538319256667337102320420, −7.406913311978162664191335956916, −6.90802279973153055685788275485, −6.356424877498956090821727967656, −5.82930983238305460122898301047, −4.77744561790866549595411197824, −3.975653608999067318486151700389, −3.59129679132829482250960619316, −2.51472689459159546080244133723, −1.2693248520158756662670038117, −0.027267563830390513208124359570,
0.461517757536513389975835006753, 1.314505467659902466245660972401, 2.59721302322791308954585290149, 3.56978907983642374289038036315, 3.90508716583646349358135419279, 4.7089828450006731470810924818, 5.50505700671383786881967897596, 6.02125101340342485186046189906, 6.96820952634736634664247283284, 7.62696774892896619555813240467, 9.12123720984174306421249123293, 9.41109372390749640202513664547, 10.15089779984289683719210205250, 10.9037374447420650243201926546, 11.66775624983932350915046552554, 12.10181255033378528150445909687, 12.54299163650034082198872237788, 13.318241717323064009042241590014, 14.14162530614756776903189256743, 15.23271070796902258333085839402, 15.48286967188124400657922891670, 16.118400823587910495489240341221, 16.84843333666496271767934941831, 17.697377081050769230337346582638, 18.47747083013102579421054554716