L(s) = 1 | + (−0.786 − 0.618i)2-s + (−0.654 − 0.755i)3-s + (0.235 + 0.971i)4-s + (0.841 + 0.540i)5-s + (0.0475 + 0.998i)6-s + (0.928 − 0.371i)7-s + (0.415 − 0.909i)8-s + (−0.142 + 0.989i)9-s + (−0.327 − 0.945i)10-s + (0.0475 − 0.998i)11-s + (0.580 − 0.814i)12-s + (−0.995 − 0.0950i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (−0.888 + 0.458i)16-s + (0.235 − 0.971i)17-s + ⋯ |
L(s) = 1 | + (−0.786 − 0.618i)2-s + (−0.654 − 0.755i)3-s + (0.235 + 0.971i)4-s + (0.841 + 0.540i)5-s + (0.0475 + 0.998i)6-s + (0.928 − 0.371i)7-s + (0.415 − 0.909i)8-s + (−0.142 + 0.989i)9-s + (−0.327 − 0.945i)10-s + (0.0475 − 0.998i)11-s + (0.580 − 0.814i)12-s + (−0.995 − 0.0950i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (−0.888 + 0.458i)16-s + (0.235 − 0.971i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5116210080 - 0.3824134433i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5116210080 - 0.3824134433i\) |
\(L(1)\) |
\(\approx\) |
\(0.6508603162 - 0.3115735324i\) |
\(L(1)\) |
\(\approx\) |
\(0.6508603162 - 0.3115735324i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 + (-0.786 - 0.618i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.928 - 0.371i)T \) |
| 11 | \( 1 + (0.0475 - 0.998i)T \) |
| 13 | \( 1 + (-0.995 - 0.0950i)T \) |
| 17 | \( 1 + (0.235 - 0.971i)T \) |
| 19 | \( 1 + (0.928 + 0.371i)T \) |
| 23 | \( 1 + (0.981 - 0.189i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.995 + 0.0950i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.723 + 0.690i)T \) |
| 43 | \( 1 + (-0.959 + 0.281i)T \) |
| 47 | \( 1 + (-0.327 + 0.945i)T \) |
| 53 | \( 1 + (-0.959 - 0.281i)T \) |
| 59 | \( 1 + (0.415 - 0.909i)T \) |
| 61 | \( 1 + (0.0475 + 0.998i)T \) |
| 71 | \( 1 + (0.235 + 0.971i)T \) |
| 73 | \( 1 + (0.0475 + 0.998i)T \) |
| 79 | \( 1 + (0.580 - 0.814i)T \) |
| 83 | \( 1 + (-0.888 + 0.458i)T \) |
| 89 | \( 1 + (-0.654 + 0.755i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−32.888805194309940621231291840990, −31.46088487296462683791568544089, −29.58008925460400875689777280194, −28.50199533061421009425901343546, −27.9309997256781868371016524387, −26.8836862098245404582012592169, −25.72330501456284272233296006818, −24.60724951176182595125327626057, −23.685334839911422874900522137943, −22.19725699254338371440428332518, −21.05999613884446391166397862923, −20.02028162619582332396245195017, −18.13869202660954032422179554343, −17.42082952620156083898981271476, −16.67998524542815225915458986583, −15.24039651595282928092580347824, −14.452918374746916554439865098181, −12.36852962459091072852452104866, −10.916037915155044189430855362213, −9.75718964277253090260535305797, −8.917658293826800817898006415883, −7.18457258664405498497523274124, −5.5282736102232391813432103485, −4.87707164385907087732548072677, −1.73393371002523266968049104033,
1.30573367910127874042384673796, 2.77644808941050572530356814260, 5.24873633831191262004010383208, 6.92510580785494756973639825051, 7.934430775507692782876974475750, 9.6549646307402747732252962982, 10.9439128483701088388861991459, 11.670530138065214141310213461910, 13.17828453499795267148072134784, 14.25706445442116852613897594257, 16.53784655422826880189149828929, 17.37632043420776959462020239647, 18.25571070878207072397800913490, 19.07215492241585650509054547071, 20.571047551921477889068385682429, 21.71527705957596499623672500136, 22.655424212081156817979167662685, 24.39592474753478529480308692338, 25.07626100062561817942368830801, 26.658370794016340068906460829754, 27.377980862003480697580033909321, 28.877382097555655682250605230508, 29.498275236458091783266251652463, 30.19430096766526233820871946449, 31.31663363627132706445819317309