L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.973 − 0.230i)3-s + (0.766 − 0.642i)4-s + (0.893 + 0.448i)5-s + (−0.835 + 0.549i)6-s + (−0.835 + 0.549i)7-s + (−0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (0.597 − 0.802i)12-s + (−0.0581 − 0.998i)13-s + (0.597 − 0.802i)14-s + (0.973 + 0.230i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.973 − 0.230i)3-s + (0.766 − 0.642i)4-s + (0.893 + 0.448i)5-s + (−0.835 + 0.549i)6-s + (−0.835 + 0.549i)7-s + (−0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (0.597 − 0.802i)12-s + (−0.0581 − 0.998i)13-s + (0.597 − 0.802i)14-s + (0.973 + 0.230i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0675 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0675 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6438043378 - 0.6016623117i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6438043378 - 0.6016623117i\) |
\(L(1)\) |
\(\approx\) |
\(0.8658510091 + 0.03067643183i\) |
\(L(1)\) |
\(\approx\) |
\(0.8658510091 + 0.03067643183i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (0.973 - 0.230i)T \) |
| 5 | \( 1 + (0.893 + 0.448i)T \) |
| 7 | \( 1 + (-0.835 + 0.549i)T \) |
| 11 | \( 1 + (-0.993 + 0.116i)T \) |
| 13 | \( 1 + (-0.0581 - 0.998i)T \) |
| 17 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.993 - 0.116i)T \) |
| 31 | \( 1 + (-0.286 - 0.957i)T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.173 + 0.984i)T \) |
| 47 | \( 1 + (-0.0581 + 0.998i)T \) |
| 53 | \( 1 + (0.597 - 0.802i)T \) |
| 59 | \( 1 + (0.973 + 0.230i)T \) |
| 61 | \( 1 + (-0.993 - 0.116i)T \) |
| 67 | \( 1 + (0.396 - 0.918i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 + (0.396 - 0.918i)T \) |
| 79 | \( 1 + (-0.993 + 0.116i)T \) |
| 83 | \( 1 + (0.893 + 0.448i)T \) |
| 89 | \( 1 + (-0.286 + 0.957i)T \) |
| 97 | \( 1 + (0.973 - 0.230i)T \) |
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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
−18.653143420099740956667182647746, −18.1119635565232076235465152028, −17.22881398810010892439139099720, −16.68899339782710012523455572349, −15.90049566785680080152356794011, −15.60468764399083932225452476538, −14.53457061262212629889358623275, −13.5449418051082515468647079271, −13.23427796111993778406393491575, −12.74887971220574590304479764404, −11.58308910682734844186772746281, −10.72660231510292475963730253158, −10.16867474698115976446377743706, −9.54769683618734482694011376170, −8.92013743284548966805864362705, −8.647033101816374538366495522156, −7.36709433000758538471001907058, −7.092970662591695568323913880866, −6.20235660829846072930866211456, −5.06458180329221823510455540630, −4.17646028028608740819334843168, −3.34346856361985990832321868634, −2.503035429981222443320192107945, −2.04499921253571053276306177087, −1.08179510528628425491357536870,
0.275985858140029864222712200173, 1.66692597188040252815288093111, 2.42228845489502583009180156755, 2.68871125408616239387374057746, 3.7007631068321939330442897727, 5.074030426094625638526486070234, 6.01442590823069848314758409037, 6.32533774851626928256091402857, 7.28443707398928228436091953555, 7.88115080699887687237973923972, 8.64154178701364287511366952877, 9.23521826968024605079150250411, 9.90827123697646357151325690518, 10.41077525740406914335445474983, 11.019901789596695143438106580802, 12.40313538888614648382253778211, 12.94234368239988781063562735654, 13.45961501054815431562591753295, 14.52460454225195231123334939168, 14.99748368159458456703515490563, 15.48386392218659620578847528626, 16.247492947702142987838573741269, 17.024624978422131029989898707127, 17.967237336857393524362856701, 18.24205067498872588094019832444