L(s) = 1 | + (0.207 + 0.978i)2-s + (0.866 + 0.5i)3-s + (−0.913 + 0.406i)4-s + (−0.309 + 0.951i)6-s + (−0.587 − 0.809i)8-s + (0.5 + 0.866i)9-s + (−0.994 − 0.104i)12-s − i·13-s + (0.669 − 0.743i)16-s + (−0.743 + 0.669i)17-s + (−0.743 + 0.669i)18-s + (0.913 + 0.406i)19-s + (−0.406 + 0.913i)23-s + (−0.104 − 0.994i)24-s + (0.978 − 0.207i)26-s + i·27-s + ⋯ |
L(s) = 1 | + (0.207 + 0.978i)2-s + (0.866 + 0.5i)3-s + (−0.913 + 0.406i)4-s + (−0.309 + 0.951i)6-s + (−0.587 − 0.809i)8-s + (0.5 + 0.866i)9-s + (−0.994 − 0.104i)12-s − i·13-s + (0.669 − 0.743i)16-s + (−0.743 + 0.669i)17-s + (−0.743 + 0.669i)18-s + (0.913 + 0.406i)19-s + (−0.406 + 0.913i)23-s + (−0.104 − 0.994i)24-s + (0.978 − 0.207i)26-s + i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1037520597 + 1.890039431i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1037520597 + 1.890039431i\) |
\(L(1)\) |
\(\approx\) |
\(0.9280558161 + 0.9814432559i\) |
\(L(1)\) |
\(\approx\) |
\(0.9280558161 + 0.9814432559i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.207 + 0.978i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 + (-0.743 + 0.669i)T \) |
| 19 | \( 1 + (0.913 + 0.406i)T \) |
| 23 | \( 1 + (-0.406 + 0.913i)T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.913 + 0.406i)T \) |
| 37 | \( 1 + (0.406 - 0.913i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.207 - 0.978i)T \) |
| 59 | \( 1 + (-0.104 - 0.994i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.207 + 0.978i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.207 + 0.978i)T \) |
| 79 | \( 1 + (-0.669 + 0.743i)T \) |
| 83 | \( 1 + (-0.951 - 0.309i)T \) |
| 89 | \( 1 + (0.913 + 0.406i)T \) |
| 97 | \( 1 + (-0.951 + 0.309i)T \) |
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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
−19.81117088594578071286660867278, −19.007591172507007541453027752401, −18.48064023458066101021066435216, −17.90538138465367706995371886992, −16.95170718719834941864364842343, −15.81725856193477409604582633881, −15.08125331498284240799713432667, −14.106159319659876080600227697149, −13.81564088314190298774845466468, −13.13801225037335537663548152492, −12.137165322567058088661327592994, −11.79192647056365249435568231139, −10.78781267295350818231243030367, −9.88901474982765394237202177170, −9.11940720458665680133757531680, −8.73699908752930677789194570600, −7.66445091866527892283490465034, −6.8338708868277549550322368353, −5.906122252497717411106615270663, −4.66917332561579919176474758260, −4.11571273928624678281108362967, −3.09455403444286389685497245104, −2.40310778676768066411627765130, −1.67363397858922194616241185704, −0.55772155800217477509235544260,
1.32096179853214973888828453245, 2.69084543166953081312875687261, 3.51481609280689224858907689737, 4.15653379816293862993172246724, 5.16472595892635755888780699553, 5.73737588459429945264485516999, 6.8879452884915813763995534785, 7.642526975599482598600494631391, 8.27312238458212133153262573641, 8.9398337414318834914257589418, 9.77781512122366458906936742030, 10.37915253337800359763553848200, 11.49615264011998052843797487018, 12.74057773850252096180368859220, 13.12766082786597023537692007383, 14.01168095850245554333789250979, 14.610970545230049343979136877164, 15.2336861156393061746823287319, 15.99020362457629866442922541736, 16.34084928148432715563778899681, 17.542613323160265028125398284867, 17.9674036685606677597996334900, 18.875729257193755223377895053097, 19.843235937617636994991380714215, 20.22487423621575382465540830306