L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + 12-s + (0.309 − 0.951i)14-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s + (−0.309 + 0.951i)19-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + 12-s + (0.309 − 0.951i)14-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s + (−0.309 + 0.951i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.886222678 - 0.2546613688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.886222678 - 0.2546613688i\) |
\(L(1)\) |
\(\approx\) |
\(1.740397797 - 0.08475182329i\) |
\(L(1)\) |
\(\approx\) |
\(1.740397797 - 0.08475182329i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.809 - 0.587i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (-0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.309 - 0.951i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−28.562241347997928214835600274729, −27.55359342875669049627740574432, −26.329732580839071912328767641188, −25.33725907786765514275419549909, −24.51669145561170883530686484285, −22.81095985878247945016113593174, −22.25004071505372329824537034453, −21.45914909563681067343503407608, −20.768270068660098383698944656202, −19.5263136506946573413251884844, −18.61766597263153108968646225930, −17.229466198561090311988604620076, −15.56406987532350740389740381954, −15.203291644527540898314708397381, −13.9796666714458545311860303451, −13.19838285861419905611523847788, −11.675142305110714200529580662770, −10.76442034782587981517762021830, −9.70481427945259393806950526712, −8.94042141425790423504054578283, −6.6653256225338118016092549232, −5.58425473385700245338895090393, −4.557639638626947876015963048870, −2.97867548231672152663672141224, −2.35893887996558478099649173383,
1.61470811451657133080588744696, 3.13587534350305015869331114231, 4.603868496788767810135624728351, 6.03764479234011000039813530056, 6.81028382342719733528711473904, 8.00165927775229704384080728999, 9.07020779707982328636487843990, 10.79370453294874704819507544046, 12.40724852679420271993591586352, 13.01827271199565210377125477513, 13.83931300602526806918183004162, 14.64091058631613377894482750044, 16.21029779299586599570857229003, 17.17553295201078738476341406869, 17.81391162826051428011592394415, 19.44206990944816401795606610524, 20.441681101143701227511984724205, 21.26257220450938616910408089382, 22.62508239731322414107668540562, 23.49594348140967702004220293087, 24.31968475728305822734074417547, 25.149161596629434905546759366335, 25.8680079397974037753439840523, 26.86394583188589995033715201734, 28.77689906556051991042912394175