L(s) = 1 | + (−0.361 + 0.932i)2-s + (0.850 + 0.526i)3-s + (−0.739 − 0.673i)4-s + (−0.602 − 0.798i)5-s + (−0.798 + 0.602i)6-s + (−0.183 + 0.982i)7-s + (0.895 − 0.445i)8-s + (0.445 + 0.895i)9-s + (0.961 − 0.273i)10-s + (−0.273 + 0.961i)11-s + (−0.273 − 0.961i)12-s + (0.361 + 0.932i)13-s + (−0.850 − 0.526i)14-s + (−0.0922 − 0.995i)15-s + (0.0922 + 0.995i)16-s + (0.602 + 0.798i)17-s + ⋯ |
L(s) = 1 | + (−0.361 + 0.932i)2-s + (0.850 + 0.526i)3-s + (−0.739 − 0.673i)4-s + (−0.602 − 0.798i)5-s + (−0.798 + 0.602i)6-s + (−0.183 + 0.982i)7-s + (0.895 − 0.445i)8-s + (0.445 + 0.895i)9-s + (0.961 − 0.273i)10-s + (−0.273 + 0.961i)11-s + (−0.273 − 0.961i)12-s + (0.361 + 0.932i)13-s + (−0.850 − 0.526i)14-s + (−0.0922 − 0.995i)15-s + (0.0922 + 0.995i)16-s + (0.602 + 0.798i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02606118788 + 2.034768816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02606118788 + 2.034768816i\) |
\(L(1)\) |
\(\approx\) |
\(0.7635873283 + 0.7845942207i\) |
\(L(1)\) |
\(\approx\) |
\(0.7635873283 + 0.7845942207i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (-0.361 + 0.932i)T \) |
| 3 | \( 1 + (0.850 + 0.526i)T \) |
| 5 | \( 1 + (-0.602 - 0.798i)T \) |
| 7 | \( 1 + (-0.183 + 0.982i)T \) |
| 11 | \( 1 + (-0.273 + 0.961i)T \) |
| 13 | \( 1 + (0.361 + 0.932i)T \) |
| 17 | \( 1 + (0.602 + 0.798i)T \) |
| 19 | \( 1 + (0.895 + 0.445i)T \) |
| 23 | \( 1 + (0.0922 - 0.995i)T \) |
| 29 | \( 1 + (0.982 - 0.183i)T \) |
| 31 | \( 1 + (0.895 - 0.445i)T \) |
| 37 | \( 1 + (0.526 - 0.850i)T \) |
| 41 | \( 1 + (-0.445 + 0.895i)T \) |
| 43 | \( 1 + (-0.183 + 0.982i)T \) |
| 47 | \( 1 + (-0.850 - 0.526i)T \) |
| 53 | \( 1 + (0.995 + 0.0922i)T \) |
| 59 | \( 1 + (0.526 - 0.850i)T \) |
| 61 | \( 1 + (0.850 + 0.526i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.445 + 0.895i)T \) |
| 73 | \( 1 + (-0.850 + 0.526i)T \) |
| 79 | \( 1 + (0.273 + 0.961i)T \) |
| 83 | \( 1 + (0.850 + 0.526i)T \) |
| 89 | \( 1 + (0.739 - 0.673i)T \) |
| 97 | \( 1 + (0.183 - 0.982i)T \) |
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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
−20.69136448995645544368621639784, −20.14442856718753784568796555832, −19.42327246501253375333439717123, −18.937749715299198410546408768337, −18.10148838990861166620728284226, −17.5640477488567667807160784515, −16.25565278679530473671654250370, −15.539540641318146810941176839278, −14.32469523511592229802523297947, −13.647679186186781139071463737059, −13.32411310804531851912398893480, −12.06207380805688926877012780073, −11.47580389346488725400779377463, −10.455715232788235216204246315778, −9.96729572059675590372659390311, −8.83718477739149985239339045334, −7.93382593462846002801796003432, −7.535762330324119345859693068719, −6.59213332654935905093913106199, −5.05151249052629296838713721283, −3.65198803445437497227785206620, −3.30231128465071562084085597164, −2.64121000736412553233963311283, −1.066305096865614620610246640128, −0.55041308264860046359310598605,
1.17538436071557729333904922473, 2.284051276393303451813206384171, 3.70000675014556560057741955197, 4.53828911016225574229860235656, 5.17676918731171316922714949495, 6.289749730271386260156950068537, 7.37947364625269908716491884755, 8.37245153861090058237937839349, 8.50412729334133865376410536183, 9.68716913445449638641512888042, 9.93660101106512373739212248033, 11.43581228173242663033745766003, 12.49548199327558537335140589878, 13.1685832840882095961777307857, 14.28030399806075624022181522152, 14.88508499505288480416457367664, 15.58135930333848740676792693840, 16.20984107298938677772949013239, 16.713114678550899971806824981456, 17.94155825811621876921168057399, 18.75580818977107138889571260574, 19.3566247636432660247691906844, 20.11318229189319350289199394621, 21.01527360418583048741278448204, 21.694985400524734187607516183761