L(s) = 1 | + (−0.0186 − 0.999i)2-s + (0.992 + 0.123i)3-s + (−0.999 + 0.0372i)4-s + (−0.926 + 0.375i)5-s + (0.105 − 0.994i)6-s + (−0.820 − 0.571i)7-s + (0.0558 + 0.998i)8-s + (0.969 + 0.245i)9-s + (0.392 + 0.919i)10-s + (0.998 − 0.0496i)11-s + (−0.996 − 0.0868i)12-s + (−0.635 + 0.771i)13-s + (−0.556 + 0.831i)14-s + (−0.966 + 0.257i)15-s + (0.997 − 0.0744i)16-s + (−0.166 − 0.985i)17-s + ⋯ |
L(s) = 1 | + (−0.0186 − 0.999i)2-s + (0.992 + 0.123i)3-s + (−0.999 + 0.0372i)4-s + (−0.926 + 0.375i)5-s + (0.105 − 0.994i)6-s + (−0.820 − 0.571i)7-s + (0.0558 + 0.998i)8-s + (0.969 + 0.245i)9-s + (0.392 + 0.919i)10-s + (0.998 − 0.0496i)11-s + (−0.996 − 0.0868i)12-s + (−0.635 + 0.771i)13-s + (−0.556 + 0.831i)14-s + (−0.966 + 0.257i)15-s + (0.997 − 0.0744i)16-s + (−0.166 − 0.985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9822573983 - 0.9491472552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9822573983 - 0.9491472552i\) |
\(L(1)\) |
\(\approx\) |
\(0.9923759344 - 0.5212010222i\) |
\(L(1)\) |
\(\approx\) |
\(0.9923759344 - 0.5212010222i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.0186 - 0.999i)T \) |
| 3 | \( 1 + (0.992 + 0.123i)T \) |
| 5 | \( 1 + (-0.926 + 0.375i)T \) |
| 7 | \( 1 + (-0.820 - 0.571i)T \) |
| 11 | \( 1 + (0.998 - 0.0496i)T \) |
| 13 | \( 1 + (-0.635 + 0.771i)T \) |
| 17 | \( 1 + (-0.166 - 0.985i)T \) |
| 19 | \( 1 + (0.901 - 0.432i)T \) |
| 29 | \( 1 + (0.369 - 0.929i)T \) |
| 31 | \( 1 + (0.988 + 0.148i)T \) |
| 37 | \( 1 + (-0.596 - 0.802i)T \) |
| 41 | \( 1 + (0.664 + 0.747i)T \) |
| 43 | \( 1 + (0.566 - 0.824i)T \) |
| 47 | \( 1 + (-0.990 - 0.136i)T \) |
| 53 | \( 1 + (0.392 - 0.919i)T \) |
| 59 | \( 1 + (-0.263 - 0.964i)T \) |
| 61 | \( 1 + (0.783 - 0.621i)T \) |
| 67 | \( 1 + (-0.616 + 0.787i)T \) |
| 71 | \( 1 + (0.645 + 0.763i)T \) |
| 73 | \( 1 + (0.369 + 0.929i)T \) |
| 79 | \( 1 + (-0.514 - 0.857i)T \) |
| 83 | \( 1 + (-0.0434 + 0.999i)T \) |
| 89 | \( 1 + (-0.471 - 0.882i)T \) |
| 97 | \( 1 + (0.955 + 0.293i)T \) |
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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
−24.1287168611805744690048233898, −22.796284094583299383005284366170, −22.358613898552606652436721853860, −21.25478293153394660033548557357, −19.89239828058034892304563938998, −19.52942306840012846133330288895, −18.7807501118332246838360605461, −17.7384052422929471460460446904, −16.65663261155570017336847810850, −15.86125520770678750437615841625, −15.19180508726060662389765065058, −14.59833410990215844362941940436, −13.55745983849548175350039107982, −12.56928130165008688483396251151, −12.14634688846715019674349537791, −10.24449986708200453577559024327, −9.340623929860826179745407567, −8.65631136714634851630557315971, −7.88137664878244671434492973070, −7.05143396316824790355506328803, −6.0923130275718593149407524393, −4.792253752107219438131233115987, −3.79015585143799066928735440998, −3.04989940708853425807621821411, −1.15657974925221627197318457339,
0.818755788942825180109597472915, 2.37255029640200535509561387992, 3.234930506851742184585425445903, 3.9787684409104737183792017232, 4.71781837718642329394336462084, 6.74875298660938283097883911212, 7.47250928016171483318449800137, 8.60077567823360768788995783817, 9.51473720682197059617140144544, 9.99405541694957708835895709268, 11.29626495499995270652641807538, 11.94042869463054307947401341914, 12.93574557706355812433476264216, 14.02321967242702037916403037351, 14.28144433551607129114758921356, 15.57295007441059778398962039035, 16.37523713998856945191780019791, 17.570658246525616221592807323006, 18.76696008568067518422925752449, 19.36463034130319538673588162207, 19.7857835398985143304690324564, 20.48510375059663429722012721357, 21.51321891816974430558083456680, 22.443973407529952553942995826567, 22.89483639293317997065758787923