L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s − 7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.309 + 0.951i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s − 18-s + (−0.809 − 0.587i)19-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s − 7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.309 + 0.951i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s − 18-s + (−0.809 − 0.587i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5240838236 - 0.4335598446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5240838236 - 0.4335598446i\) |
\(L(1)\) |
\(\approx\) |
\(0.7706990758 - 0.4515038146i\) |
\(L(1)\) |
\(\approx\) |
\(0.7706990758 - 0.4515038146i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.809 + 0.587i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 + (0.809 + 0.587i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.809 - 0.587i)T \) |
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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
−38.36385625028819045647228142415, −37.26380364511621541164631254905, −36.11669324804437190080436291140, −34.91139412540518781839425671576, −33.51027636896173048414152156095, −32.142785570811738917279883934617, −31.863185453142629020134564875139, −29.72059632287232218730806481137, −27.81613798359506846716938392087, −26.851369901539873776928623922210, −25.647646736090431891325639246723, −24.83111727710086347653701125163, −23.02881841425419766842900148042, −21.74419316688276455445293419541, −19.815615669641949688948559710507, −18.779510872615321987949064088151, −16.77026179013987180530517255610, −15.7750222662889525951832674177, −14.47471425031826237727039410793, −13.17474937252591201691648066386, −10.2649821979982712510127350133, −9.1007059557284357333270928515, −7.65721871714893782562673106625, −5.69525797854698655509075110454, −3.56286301215718626248462418835,
2.13858987186166861971624060558, 3.87162723584480278913458272457, 7.0252574621156174444793516646, 8.83004689737836649629298973739, 9.984614757569034276556628636003, 12.15610361504569240431112275340, 13.108533409221915032416646344925, 14.64851488374259577660623929047, 16.92435450015065411443429305569, 18.59221496591904323290261617352, 19.48026988227819791926043054632, 20.560065527476893369493659005480, 22.0873815233214598442836214081, 23.61413591520153963594981421375, 25.58333562770774913914970287131, 26.28598245387479879441431448339, 28.03366939887295734630480981476, 29.29868373787037724456850028721, 30.345757362397796713437597878831, 31.45895811645762387438683159672, 32.53423775786614196310832036417, 34.887120850584521875781093352581, 36.05393354849720361244723971653, 36.692984174110792567464581601384, 38.28063243031528312009180392797