L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)6-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 − 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + 18-s + (0.939 + 0.342i)21-s + (0.173 + 0.984i)22-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)6-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 − 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + 18-s + (0.939 + 0.342i)21-s + (0.173 + 0.984i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9290411688 - 0.4463925469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9290411688 - 0.4463925469i\) |
\(L(1)\) |
\(\approx\) |
\(0.7771671808 - 0.07758658242i\) |
\(L(1)\) |
\(\approx\) |
\(0.7771671808 - 0.07758658242i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (0.939 + 0.342i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.939 - 0.342i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.766 + 0.642i)T \) |
| 59 | \( 1 + (0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.766 + 0.642i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.173 + 0.984i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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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
−29.989769428922358878798569971523, −28.731409640822660768760028931719, −28.23946018200689080215448421191, −26.9036832230336598162581584719, −25.57962055758632010284312028012, −25.2076205649955216449185658736, −23.95930950640067444227095785474, −23.30312576322559787046777067785, −21.37308051862811368514996714264, −20.23379661507116250908859486634, −19.12081535647369985991326849522, −18.2876176287608750384917585599, −17.62560814567443355266868707280, −16.248033565916484488703733562789, −14.98597419640962542176289683363, −14.01001825406222704214576979769, −12.27010191446990908290445773917, −11.5242735982664726238167862903, −9.84694199342622915256678593237, −8.633407191238348213662731911923, −7.7048481772197433553379445002, −6.58502890864581983620781618586, −5.3067891655173060757304439237, −2.54467343083316768704561601172, −1.42550695400141056749037600792,
0.65866254621464409718261893224, 2.808465329216085950946789462173, 4.04350697501726152293043564885, 5.82557344806491324866721551023, 7.795482352239347116468203307510, 8.50227447118137286522556636307, 10.12212972520178229595769097618, 10.55771622041542564414227818994, 11.75638334919090956413217047778, 13.474686770567993444519672079781, 14.87352271757780502359869290567, 16.0850559689298936402092553226, 16.86825426808500863446325516036, 17.937234665844706126613919781770, 19.28884272461440731228198556858, 20.388437003004347270405358938504, 20.979949829298519251432116789405, 22.055797183732882101645115012850, 23.50099123303605390131785330194, 24.92131738571691288222474252710, 26.05952177020344839207985079112, 26.80235153071861562680056517873, 27.555885661174959442862675487723, 28.46942503005971554000651886550, 29.71880757382283544365850611438