L(s) = 1 | + (−0.453 + 0.891i)3-s + i·7-s + (−0.587 − 0.809i)9-s + (−0.987 + 0.156i)11-s + (−0.987 − 0.156i)13-s + (−0.309 + 0.951i)17-s + (−0.891 + 0.453i)19-s + (−0.891 − 0.453i)21-s + (−0.587 + 0.809i)23-s + (0.987 − 0.156i)27-s + (0.453 − 0.891i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.156 + 0.987i)37-s + (0.587 − 0.809i)39-s + ⋯ |
L(s) = 1 | + (−0.453 + 0.891i)3-s + i·7-s + (−0.587 − 0.809i)9-s + (−0.987 + 0.156i)11-s + (−0.987 − 0.156i)13-s + (−0.309 + 0.951i)17-s + (−0.891 + 0.453i)19-s + (−0.891 − 0.453i)21-s + (−0.587 + 0.809i)23-s + (0.987 − 0.156i)27-s + (0.453 − 0.891i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.156 + 0.987i)37-s + (0.587 − 0.809i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09446482433 - 0.03611984500i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09446482433 - 0.03611984500i\) |
\(L(1)\) |
\(\approx\) |
\(0.5669052557 + 0.3085304511i\) |
\(L(1)\) |
\(\approx\) |
\(0.5669052557 + 0.3085304511i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.453 + 0.891i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-0.987 + 0.156i)T \) |
| 13 | \( 1 + (-0.987 - 0.156i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.891 + 0.453i)T \) |
| 23 | \( 1 + (-0.587 + 0.809i)T \) |
| 29 | \( 1 + (0.453 - 0.891i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.156 + 0.987i)T \) |
| 41 | \( 1 + (-0.587 - 0.809i)T \) |
| 43 | \( 1 + (0.707 - 0.707i)T \) |
| 47 | \( 1 + (0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.891 - 0.453i)T \) |
| 59 | \( 1 + (0.156 - 0.987i)T \) |
| 61 | \( 1 + (0.156 + 0.987i)T \) |
| 67 | \( 1 + (0.891 - 0.453i)T \) |
| 71 | \( 1 + (-0.951 + 0.309i)T \) |
| 73 | \( 1 + (0.587 - 0.809i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.891 + 0.453i)T \) |
| 89 | \( 1 + (-0.587 + 0.809i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−22.3234619446534407019428853026, −21.4421327517679376434409113744, −20.308602296031796963938974553853, −19.80495501628424556632156245438, −18.85320028701907723635074344278, −18.13712099712839680274249540045, −17.38220424851135865836118906383, −16.65063373581824179712784567566, −15.937792666785725675681784952541, −14.625806207170301494971284227463, −13.911801035392161754699345015811, −13.07830105561818572008683806110, −12.51780488054116488219762318993, −11.43141647740325401250764315059, −10.73866458885861362625167353964, −9.93710644633029271287560304022, −8.65014664185456752600889465898, −7.65882498848086024484567545250, −7.13556290762341912437440637531, −6.261420955397134916697833190410, −5.11359216381377005187645044013, −4.40140508155365080718233582225, −2.84604521596373992979040206011, −2.04779217838404424336821728861, −0.621005135125986619800976193730,
0.03661871467776540475939760042, 1.95792371569643903145955879354, 2.909052543372982878060015850265, 4.0492736380116036088775437480, 5.03290812590734130086788897909, 5.66695758188001742671619149096, 6.559348728934825810327203763056, 7.948786308129724187159529647994, 8.68307561137136465768633665477, 9.7244770539089831204335302500, 10.32197271612603942939468600990, 11.18259221335258553877183591878, 12.22637314622450736149356927718, 12.63880990690634368343689360843, 13.99875910360242485227929673513, 15.05022579080250518647138384327, 15.40773336729113619631990016535, 16.16888973553338628949263142757, 17.30156800821147008828408228573, 17.6639239952527967265672948860, 18.807265975919353181165414683, 19.53775299252889891598625559905, 20.631583437401330787415499896539, 21.313455378618477506605514253290, 21.92428472738511372452923445785