L(s) = 1 | + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)6-s + (−0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (0.913 + 0.406i)11-s + (0.913 − 0.406i)12-s + (−0.809 − 0.587i)13-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (0.5 + 0.866i)18-s + (0.978 + 0.207i)19-s + (−0.309 + 0.951i)22-s + (0.104 + 0.994i)23-s + (0.5 + 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)6-s + (−0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (0.913 + 0.406i)11-s + (0.913 − 0.406i)12-s + (−0.809 − 0.587i)13-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (0.5 + 0.866i)18-s + (0.978 + 0.207i)19-s + (−0.309 + 0.951i)22-s + (0.104 + 0.994i)23-s + (0.5 + 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6408245378 + 0.9656563767i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6408245378 + 0.9656563767i\) |
\(L(1)\) |
\(\approx\) |
\(0.6843793783 + 0.4614661394i\) |
\(L(1)\) |
\(\approx\) |
\(0.6843793783 + 0.4614661394i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 11 | \( 1 + (0.913 + 0.406i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (0.978 + 0.207i)T \) |
| 23 | \( 1 + (0.104 + 0.994i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.913 + 0.406i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.669 + 0.743i)T \) |
| 53 | \( 1 + (0.978 - 0.207i)T \) |
| 59 | \( 1 + (0.104 - 0.994i)T \) |
| 61 | \( 1 + (0.104 + 0.994i)T \) |
| 67 | \( 1 + (-0.669 + 0.743i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.913 + 0.406i)T \) |
| 79 | \( 1 + (0.669 + 0.743i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (0.104 + 0.994i)T \) |
| 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
−27.2625879144536942188781433977, −26.326621506314405294162992286172, −24.59471055946933885510985557468, −23.87338810117412444906974582569, −22.76935275439530726240730175839, −22.048277869421071314658310959568, −21.36837798641551642645425903396, −20.02217892003999177805520847029, −19.09797525966795037654014349111, −18.28631435188751758115453272485, −17.18624559864900232381523684804, −16.48367160383432176671749904875, −14.75427719290869731148286558246, −13.770407689073853321607177710933, −12.47977190740351363339972710025, −11.93081243085287859243057537787, −10.92858430129044319129838345487, −9.96831881498723940097177547664, −8.84363229621183182695277615427, −7.23057195659660161535006681318, −5.882607208873193123630091873878, −4.806908226701130301516198528259, −3.62704001019302029202415051749, −1.89890157816091942696394822316, −0.65015434292404332810958888893,
0.95378570221037474978470123060, 3.56141588436997039435408074869, 4.87024299763742503397300931044, 5.63306673981735830316593653560, 6.86606124446776009840993934313, 7.66619403348458555716760348847, 9.366334141003061673015697684457, 10.03891454916207922664669572172, 11.712640160306578983800133461119, 12.45364407555214638015147779495, 13.774967598015436172195676929832, 14.87421420659447379292389787960, 15.80558140682091713493513249134, 16.73193560968362100144117338893, 17.5097086520894257382942102932, 18.25565412595590068360512327730, 19.55435629364718219610639514248, 21.07839837649839993097936327966, 22.21746400195263231889260758021, 22.69062862268920662579308581289, 23.61442804401579731563817912234, 24.66825919552778302913651996642, 25.30577835726120420953913810466, 26.769692817824609734279903562912, 27.338654559099707973568548357078