L(s) = 1 | + (0.730 + 0.683i)2-s + (−0.743 − 0.669i)3-s + (0.0665 + 0.997i)4-s + (−0.0855 − 0.996i)6-s + (−0.633 + 0.774i)8-s + (0.104 + 0.994i)9-s + (0.618 − 0.786i)12-s + (−0.884 + 0.466i)13-s + (−0.991 + 0.132i)16-s + (−0.318 − 0.948i)17-s + (−0.603 + 0.797i)18-s + (−0.595 + 0.803i)19-s + (0.690 + 0.723i)23-s + (0.988 − 0.151i)24-s + (−0.964 − 0.263i)26-s + (0.587 − 0.809i)27-s + ⋯ |
L(s) = 1 | + (0.730 + 0.683i)2-s + (−0.743 − 0.669i)3-s + (0.0665 + 0.997i)4-s + (−0.0855 − 0.996i)6-s + (−0.633 + 0.774i)8-s + (0.104 + 0.994i)9-s + (0.618 − 0.786i)12-s + (−0.884 + 0.466i)13-s + (−0.991 + 0.132i)16-s + (−0.318 − 0.948i)17-s + (−0.603 + 0.797i)18-s + (−0.595 + 0.803i)19-s + (0.690 + 0.723i)23-s + (0.988 − 0.151i)24-s + (−0.964 − 0.263i)26-s + (0.587 − 0.809i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1417654541 + 0.3111996361i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1417654541 + 0.3111996361i\) |
\(L(1)\) |
\(\approx\) |
\(0.8682481765 + 0.3837927733i\) |
\(L(1)\) |
\(\approx\) |
\(0.8682481765 + 0.3837927733i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.730 + 0.683i)T \) |
| 3 | \( 1 + (-0.743 - 0.669i)T \) |
| 13 | \( 1 + (-0.884 + 0.466i)T \) |
| 17 | \( 1 + (-0.318 - 0.948i)T \) |
| 19 | \( 1 + (-0.595 + 0.803i)T \) |
| 23 | \( 1 + (0.690 + 0.723i)T \) |
| 29 | \( 1 + (0.736 + 0.676i)T \) |
| 31 | \( 1 + (-0.272 + 0.962i)T \) |
| 37 | \( 1 + (-0.768 - 0.640i)T \) |
| 41 | \( 1 + (-0.516 + 0.856i)T \) |
| 43 | \( 1 + (0.540 + 0.841i)T \) |
| 47 | \( 1 + (0.603 + 0.797i)T \) |
| 53 | \( 1 + (0.132 - 0.991i)T \) |
| 59 | \( 1 + (0.483 - 0.875i)T \) |
| 61 | \( 1 + (0.290 - 0.956i)T \) |
| 67 | \( 1 + (0.945 + 0.327i)T \) |
| 71 | \( 1 + (-0.870 + 0.491i)T \) |
| 73 | \( 1 + (0.0760 - 0.997i)T \) |
| 79 | \( 1 + (-0.449 + 0.893i)T \) |
| 83 | \( 1 + (0.336 - 0.941i)T \) |
| 89 | \( 1 + (-0.995 - 0.0950i)T \) |
| 97 | \( 1 + (-0.931 - 0.362i)T \) |
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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
−17.8143816026720276930256305065, −17.25391704036711018010639263305, −16.67767858579047570098031151165, −15.56669886804069622633542973234, −15.215748104156305280313125266742, −14.75374833737894019788435679735, −13.75020099630649457712209752401, −13.043988614847111542766523147475, −12.32830252989697215937962518174, −11.909283741092556376283721678403, −11.01924453747517898029086142135, −10.498998601707402637656907358987, −10.06248177101090498477899450882, −9.16038928864679986546284177904, −8.513615460833921375796345297380, −7.11920819344216817683757609028, −6.564850000695792307185593415536, −5.69479208710323434093196614212, −5.19019782720378496771943479908, −4.3273598586565182901022117938, −3.97084079188702034116697824735, −2.85816502005204175000162546612, −2.24514243207094125319766222292, −1.013977769991356331568569368963, −0.087115838517446204319055395068,
1.39380273698314470702876027027, 2.32066076723864505923484524292, 3.108917327091691751287769433795, 4.16248619035744490369682706574, 5.02706841020181885716680131254, 5.28711432573554368822913436754, 6.34006474890571724843098104410, 6.8579584703309155471727055733, 7.38664848611336329568364593269, 8.14118999752100388551161976524, 8.95172768523275362418789990389, 9.86994935075006391436488299423, 10.89986948792694510574918262148, 11.49324599296704739330289763942, 12.19741661518307490922746310227, 12.67563362295485559811531528970, 13.30678346670014910468185509151, 14.252467514705187937664937558038, 14.41085806004913006517381620348, 15.5723723112421187602307903038, 16.15981629329365872663409808682, 16.68348660653455941413183450523, 17.48667401218149328289671811431, 17.774039947567727382159470892041, 18.68867378537408667058856800123