L(s) = 1 | + (0.263 − 0.964i)2-s + (−0.984 + 0.176i)3-s + (−0.861 − 0.508i)4-s + (0.0296 + 0.999i)5-s + (−0.0887 + 0.996i)6-s + (0.889 + 0.456i)7-s + (−0.717 + 0.696i)8-s + (0.937 − 0.348i)9-s + (0.972 + 0.234i)10-s + (−0.915 + 0.403i)11-s + (0.937 + 0.348i)12-s + (0.829 + 0.558i)13-s + (0.674 − 0.737i)14-s + (−0.205 − 0.978i)15-s + (0.482 + 0.875i)16-s + (0.992 − 0.118i)17-s + ⋯ |
L(s) = 1 | + (0.263 − 0.964i)2-s + (−0.984 + 0.176i)3-s + (−0.861 − 0.508i)4-s + (0.0296 + 0.999i)5-s + (−0.0887 + 0.996i)6-s + (0.889 + 0.456i)7-s + (−0.717 + 0.696i)8-s + (0.937 − 0.348i)9-s + (0.972 + 0.234i)10-s + (−0.915 + 0.403i)11-s + (0.937 + 0.348i)12-s + (0.829 + 0.558i)13-s + (0.674 − 0.737i)14-s + (−0.205 − 0.978i)15-s + (0.482 + 0.875i)16-s + (0.992 − 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8348328728 + 0.01164232054i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8348328728 + 0.01164232054i\) |
\(L(1)\) |
\(\approx\) |
\(0.8738478650 - 0.1363201062i\) |
\(L(1)\) |
\(\approx\) |
\(0.8738478650 - 0.1363201062i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 107 | \( 1 \) |
good | 2 | \( 1 + (0.263 - 0.964i)T \) |
| 3 | \( 1 + (-0.984 + 0.176i)T \) |
| 5 | \( 1 + (0.0296 + 0.999i)T \) |
| 7 | \( 1 + (0.889 + 0.456i)T \) |
| 11 | \( 1 + (-0.915 + 0.403i)T \) |
| 13 | \( 1 + (0.829 + 0.558i)T \) |
| 17 | \( 1 + (0.992 - 0.118i)T \) |
| 19 | \( 1 + (0.375 - 0.926i)T \) |
| 23 | \( 1 + (0.674 + 0.737i)T \) |
| 29 | \( 1 + (-0.630 + 0.776i)T \) |
| 31 | \( 1 + (-0.794 + 0.606i)T \) |
| 37 | \( 1 + (0.757 + 0.652i)T \) |
| 41 | \( 1 + (-0.320 - 0.947i)T \) |
| 43 | \( 1 + (0.0296 - 0.999i)T \) |
| 47 | \( 1 + (-0.320 + 0.947i)T \) |
| 53 | \( 1 + (0.263 + 0.964i)T \) |
| 59 | \( 1 + (-0.630 - 0.776i)T \) |
| 61 | \( 1 + (0.889 - 0.456i)T \) |
| 67 | \( 1 + (-0.717 - 0.696i)T \) |
| 71 | \( 1 + (-0.984 - 0.176i)T \) |
| 73 | \( 1 + (0.147 - 0.989i)T \) |
| 79 | \( 1 + (-0.533 - 0.845i)T \) |
| 83 | \( 1 + (0.582 - 0.812i)T \) |
| 89 | \( 1 + (-0.0887 - 0.996i)T \) |
| 97 | \( 1 + (0.972 + 0.234i)T \) |
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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
−29.66599547055154516440230610338, −28.340748702950931578376796342220, −27.59853996459979756603621993608, −26.661377495704739991489138847542, −25.13364774585220059487984248562, −24.35143929048912225227108695251, −23.50326672167994242775978970643, −22.92199905521736743063887719563, −21.33977418288544246574646365480, −20.74725250092798723502453408262, −18.60391325497190687945282224393, −17.85968057784257362664861515703, −16.66702873723200095919694368487, −16.316151608551605687160708614098, −14.90776276680272407927410601918, −13.420956511993089820095198432822, −12.74947171228173669674499647359, −11.43525438176510646009279344636, −10.0233782849603465938965792926, −8.28840686622590739105232955764, −7.6068837364758121128736834996, −5.84436986474611021394878632105, −5.24397231462147125923807393074, −4.03396391837051352960027608689, −0.99893311197333833821767737666,
1.674596444549819028620227850462, 3.28887046010257391643689082, 4.84659526816327171162608610328, 5.765803641418587985563571700068, 7.39649706211407629897315505280, 9.25178882837053935725686523062, 10.57233941663903379873694385124, 11.16352699179653475160591641396, 12.06781846190246358974005323816, 13.39793614040457656235830772416, 14.66412060572372816489441199024, 15.66553624934752494590426145309, 17.446279050525382625414167140887, 18.32075722697090029854965284298, 18.8050178184822034029161472936, 20.626545456782404856898972088065, 21.47408343410330170402742772899, 22.16933337396029790480149057260, 23.39374520195714693205208598478, 23.77577972751613793207475583865, 25.75205584448599709362845025822, 26.99181753337045554267924341836, 27.780033853283801791295845441239, 28.6615796697387668102145934976, 29.56089160139314936649387398743