L(s) = 1 | + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.981 + 0.189i)5-s + (0.580 − 0.814i)7-s + (0.654 − 0.755i)8-s + (0.841 + 0.540i)10-s + (0.0475 + 0.998i)13-s + (−0.981 + 0.189i)14-s + (−0.995 + 0.0950i)16-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.235 − 0.971i)20-s + (−0.580 − 0.814i)23-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)26-s + ⋯ |
L(s) = 1 | + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.981 + 0.189i)5-s + (0.580 − 0.814i)7-s + (0.654 − 0.755i)8-s + (0.841 + 0.540i)10-s + (0.0475 + 0.998i)13-s + (−0.981 + 0.189i)14-s + (−0.995 + 0.0950i)16-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.235 − 0.971i)20-s + (−0.580 − 0.814i)23-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.03544774184 - 0.5775148243i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.03544774184 - 0.5775148243i\) |
\(L(1)\) |
\(\approx\) |
\(0.5762513208 - 0.2630639567i\) |
\(L(1)\) |
\(\approx\) |
\(0.5762513208 - 0.2630639567i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.723 - 0.690i)T \) |
| 5 | \( 1 + (-0.981 + 0.189i)T \) |
| 7 | \( 1 + (0.580 - 0.814i)T \) |
| 13 | \( 1 + (0.0475 + 0.998i)T \) |
| 17 | \( 1 + (0.142 - 0.989i)T \) |
| 19 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-0.580 - 0.814i)T \) |
| 29 | \( 1 + (0.786 - 0.618i)T \) |
| 31 | \( 1 + (-0.888 + 0.458i)T \) |
| 37 | \( 1 + (0.841 + 0.540i)T \) |
| 41 | \( 1 + (-0.235 + 0.971i)T \) |
| 43 | \( 1 + (0.981 + 0.189i)T \) |
| 47 | \( 1 + (-0.235 - 0.971i)T \) |
| 53 | \( 1 + (-0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.723 + 0.690i)T \) |
| 61 | \( 1 + (0.235 + 0.971i)T \) |
| 67 | \( 1 + (0.235 - 0.971i)T \) |
| 71 | \( 1 + (0.142 + 0.989i)T \) |
| 73 | \( 1 + (0.415 - 0.909i)T \) |
| 79 | \( 1 + (-0.327 + 0.945i)T \) |
| 83 | \( 1 + (-0.580 + 0.814i)T \) |
| 89 | \( 1 + (0.142 - 0.989i)T \) |
| 97 | \( 1 + (0.981 + 0.189i)T \) |
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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
−21.690635976222990366903558903304, −20.57187490906551174731566327922, −19.968885030422430526434868916393, −19.124303890940893230179120570899, −18.558490202220283829648532296271, −17.721589082704092437464135293526, −17.06090998094950780509956323752, −15.98500750315679174362189599144, −15.60534285348457860715249742015, −14.78420364172150350193596657111, −14.2620720705442261772148420227, −12.80665259314820047046534206792, −12.160551816801436053047171894619, −11.12437050635910237435109919215, −10.55198937426884736918255688852, −9.441866770816388106673772905703, −8.5726930036246415287784095915, −7.950967619207385523391480039554, −7.48399534078826774167889692076, −6.08937724941683321950437629580, −5.55053050576142294643141138052, −4.53525257367011941581450047150, −3.454030954717265215784341365021, −2.06020180720373338294572831446, −1.04770787079769020212103716839,
0.200286290538432137549392892516, 1.02722094034523134982936929866, 2.26914302083304793975861303002, 3.233839364188318209248870851648, 4.29129106974505609621441166409, 4.67574316710493520037724930199, 6.64708367128679936378355496458, 7.22559205708424769822418584148, 8.025351156027021082615903027, 8.750568364910979292672490545948, 9.71040417574802509601449977172, 10.60147130391621713156064514234, 11.41535412624270088356331185964, 11.697831782379470878543851691252, 12.74523056271619497658552189337, 13.725788571667713942693327524332, 14.46296787588475494274166116213, 15.6008450378442974660069300171, 16.41084948986081994228653260191, 16.89482857175359305330877929023, 18.06755299107066193759629744912, 18.41139450730429941799327790114, 19.53358916978602591853134646838, 19.85893925555201601124924738683, 20.67716222372994432234918046649