L(s) = 1 | + (0.994 + 0.104i)2-s + (0.965 − 0.258i)3-s + (0.978 + 0.207i)4-s + (0.743 + 0.669i)5-s + (0.987 − 0.156i)6-s + (0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.669 + 0.743i)10-s + (−0.0523 + 0.998i)11-s + (0.998 − 0.0523i)12-s + (0.156 + 0.987i)13-s + (0.891 + 0.453i)15-s + (0.913 + 0.406i)16-s + (−0.998 − 0.0523i)17-s + (0.913 − 0.406i)18-s + (−0.933 + 0.358i)19-s + ⋯ |
L(s) = 1 | + (0.994 + 0.104i)2-s + (0.965 − 0.258i)3-s + (0.978 + 0.207i)4-s + (0.743 + 0.669i)5-s + (0.987 − 0.156i)6-s + (0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.669 + 0.743i)10-s + (−0.0523 + 0.998i)11-s + (0.998 − 0.0523i)12-s + (0.156 + 0.987i)13-s + (0.891 + 0.453i)15-s + (0.913 + 0.406i)16-s + (−0.998 − 0.0523i)17-s + (0.913 − 0.406i)18-s + (−0.933 + 0.358i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(5.834541935 + 1.985215509i\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.834541935 + 1.985215509i\) |
\(L(1)\) |
\(\approx\) |
\(2.997949960 + 0.5482150881i\) |
\(L(1)\) |
\(\approx\) |
\(2.997949960 + 0.5482150881i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.994 + 0.104i)T \) |
| 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 + (0.743 + 0.669i)T \) |
| 11 | \( 1 + (-0.0523 + 0.998i)T \) |
| 13 | \( 1 + (0.156 + 0.987i)T \) |
| 17 | \( 1 + (-0.998 - 0.0523i)T \) |
| 19 | \( 1 + (-0.933 + 0.358i)T \) |
| 23 | \( 1 + (0.104 - 0.994i)T \) |
| 29 | \( 1 + (0.453 - 0.891i)T \) |
| 31 | \( 1 + (-0.669 - 0.743i)T \) |
| 37 | \( 1 + (0.669 - 0.743i)T \) |
| 43 | \( 1 + (-0.587 + 0.809i)T \) |
| 47 | \( 1 + (-0.629 - 0.777i)T \) |
| 53 | \( 1 + (-0.544 - 0.838i)T \) |
| 59 | \( 1 + (0.913 - 0.406i)T \) |
| 61 | \( 1 + (0.406 - 0.913i)T \) |
| 67 | \( 1 + (0.838 - 0.544i)T \) |
| 71 | \( 1 + (-0.891 + 0.453i)T \) |
| 73 | \( 1 + (0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.258 - 0.965i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.358 - 0.933i)T \) |
| 97 | \( 1 + (0.891 + 0.453i)T \) |
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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
−25.32925567992916335996390314851, −24.26987824080375011175447518470, −23.659426351872702367171722198084, −22.04670722450528414390632028449, −21.67707557190953839862360249887, −20.77886554237575066444996215631, −20.02529034518121295619854945592, −19.321115479041018365868513972921, −17.88649557807475513544309076617, −16.61666518188055586202097932461, −15.76694168559296764214564845174, −14.94229143443765853979340302288, −13.87298563275358029731388329286, −13.2786503427090747000619885487, −12.637111005377221212344153409616, −11.07258004605455711307460119616, −10.24246600017029000900788925684, −8.9907660739495840852174396304, −8.146381106150492265988650711621, −6.74069470458458388049075003123, −5.581736774963415746107977561351, −4.654068816276338056229398160363, −3.4599461821656915876101331734, −2.49304148264805172579381717493, −1.29058926515042844058753328390,
2.00904658476596928426094270673, 2.30695985791047774983541878128, 3.78595651777408432162824859911, 4.68510763798104639369235409477, 6.40519231310275520791621573546, 6.79994275859903726010295058363, 8.016517875871954553542078208133, 9.34577269014594829299127821719, 10.34987281959601584963349374090, 11.50478086868675487121646229685, 12.8179964798626099942710047759, 13.32786179211371687329438048637, 14.50034985251100426956117367531, 14.726515454888782327110213605242, 15.84018217032165350764322255210, 17.103694681965363150421908455273, 18.222899689844635209313704274061, 19.200943697215686639752301290904, 20.18831142060253614627027839791, 21.02347475801222564315797021496, 21.67878646583024080774559354141, 22.692204561006308316308516490108, 23.589332403797391884849920385795, 24.645939022872544098242747085058, 25.25633905574900671278012803427