L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.993 − 0.116i)3-s + (−0.5 − 0.866i)4-s + (−0.686 − 0.727i)5-s + (0.597 − 0.802i)6-s + (0.973 − 0.230i)7-s + 8-s + (0.973 + 0.230i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.396 + 0.918i)12-s + (−0.686 − 0.727i)13-s + (−0.286 + 0.957i)14-s + (0.597 + 0.802i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.993 − 0.116i)3-s + (−0.5 − 0.866i)4-s + (−0.686 − 0.727i)5-s + (0.597 − 0.802i)6-s + (0.973 − 0.230i)7-s + 8-s + (0.973 + 0.230i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.396 + 0.918i)12-s + (−0.686 − 0.727i)13-s + (−0.286 + 0.957i)14-s + (0.597 + 0.802i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9167606310 - 0.2106638022i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9167606310 - 0.2106638022i\) |
\(L(1)\) |
\(\approx\) |
\(0.6592769660 + 0.03748792954i\) |
\(L(1)\) |
\(\approx\) |
\(0.6592769660 + 0.03748792954i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.993 - 0.116i)T \) |
| 5 | \( 1 + (-0.686 - 0.727i)T \) |
| 7 | \( 1 + (0.973 - 0.230i)T \) |
| 11 | \( 1 + (0.973 + 0.230i)T \) |
| 13 | \( 1 + (-0.686 - 0.727i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.286 + 0.957i)T \) |
| 31 | \( 1 + (0.893 - 0.448i)T \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (-0.939 - 0.342i)T \) |
| 47 | \( 1 + (0.597 + 0.802i)T \) |
| 53 | \( 1 + (0.597 - 0.802i)T \) |
| 59 | \( 1 + (0.597 + 0.802i)T \) |
| 61 | \( 1 + (0.893 + 0.448i)T \) |
| 67 | \( 1 + (0.396 - 0.918i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.286 - 0.957i)T \) |
| 79 | \( 1 + (-0.993 + 0.116i)T \) |
| 83 | \( 1 + (0.893 + 0.448i)T \) |
| 89 | \( 1 + (0.973 - 0.230i)T \) |
| 97 | \( 1 + (0.893 + 0.448i)T \) |
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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
−18.58187565944001114790044598049, −17.683239275887110671919099479292, −17.45756427088211223261340731352, −16.78869897596377014771575122233, −15.88829378688168591119227028487, −15.208001429342767130649681306363, −14.42120701537055880776652298561, −13.6693879936576382708990550926, −12.73553814646416867620458154857, −11.8130844005890458024040455229, −11.49853447307313389032131299146, −11.34554533763455386898125063842, −10.345034767751981841275831675093, −9.745604676221758183975216717787, −8.90735823543880228387809126983, −8.120730823737219984092915147521, −7.27358398047180808206881815155, −6.818760108638643167321050370674, −5.77302697720332532591838039402, −4.72684118457010784867082256905, −4.24265246116734065231723017789, −3.555038461153402950909522281875, −2.45310673720254146078913302705, −1.62526811437016606269612514854, −0.7902653289753400010027820490,
0.61971189678804320909882230174, 1.07536564608075304734326661852, 2.070575296020005179408244475098, 3.78931276564153608717577936275, 4.61217361562263125457904440829, 4.96304984273579580376953665628, 5.637094913981037539753357216423, 6.5973596431394569912299838009, 7.35663432824309113330569647332, 7.69912305608943655133125203433, 8.6037358820560531600106302477, 9.2713260524640036912199195228, 10.13493108079199847374108634007, 10.78907562334005524400522891481, 11.65506314473376879794413692846, 12.03829854034103485939639632761, 12.909076892545724290892476027377, 13.74813149780023837832181184188, 14.60459063617373539802403399119, 15.11403397110243921192669285288, 15.9711428427389365134715306346, 16.46220093974765971548147834859, 17.06611715170724173316354885584, 17.58905554524318064208154742163, 18.144806497547474625665343828820