L(s) = 1 | + (−0.602 + 0.798i)2-s + (0.0922 − 0.995i)3-s + (−0.273 − 0.961i)4-s + (0.932 − 0.361i)5-s + (0.739 + 0.673i)6-s + (0.445 − 0.895i)7-s + (0.932 + 0.361i)8-s + (−0.982 − 0.183i)9-s + (−0.273 + 0.961i)10-s + (0.445 − 0.895i)11-s + (−0.982 + 0.183i)12-s + (−0.982 − 0.183i)13-s + (0.445 + 0.895i)14-s + (−0.273 − 0.961i)15-s + (−0.850 + 0.526i)16-s + (0.739 + 0.673i)17-s + ⋯ |
L(s) = 1 | + (−0.602 + 0.798i)2-s + (0.0922 − 0.995i)3-s + (−0.273 − 0.961i)4-s + (0.932 − 0.361i)5-s + (0.739 + 0.673i)6-s + (0.445 − 0.895i)7-s + (0.932 + 0.361i)8-s + (−0.982 − 0.183i)9-s + (−0.273 + 0.961i)10-s + (0.445 − 0.895i)11-s + (−0.982 + 0.183i)12-s + (−0.982 − 0.183i)13-s + (0.445 + 0.895i)14-s + (−0.273 − 0.961i)15-s + (−0.850 + 0.526i)16-s + (0.739 + 0.673i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4413703621 - 0.9147623933i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4413703621 - 0.9147623933i\) |
\(L(1)\) |
\(\approx\) |
\(0.8105306992 - 0.3136154307i\) |
\(L(1)\) |
\(\approx\) |
\(0.8105306992 - 0.3136154307i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 953 | \( 1 \) |
good | 2 | \( 1 + (-0.602 + 0.798i)T \) |
| 3 | \( 1 + (0.0922 - 0.995i)T \) |
| 5 | \( 1 + (0.932 - 0.361i)T \) |
| 7 | \( 1 + (0.445 - 0.895i)T \) |
| 11 | \( 1 + (0.445 - 0.895i)T \) |
| 13 | \( 1 + (-0.982 - 0.183i)T \) |
| 17 | \( 1 + (0.739 + 0.673i)T \) |
| 19 | \( 1 + (-0.982 - 0.183i)T \) |
| 23 | \( 1 + (-0.850 - 0.526i)T \) |
| 29 | \( 1 + (0.739 + 0.673i)T \) |
| 31 | \( 1 + (-0.982 - 0.183i)T \) |
| 37 | \( 1 + (-0.602 - 0.798i)T \) |
| 41 | \( 1 + (-0.273 - 0.961i)T \) |
| 43 | \( 1 + (-0.602 - 0.798i)T \) |
| 47 | \( 1 + (0.739 - 0.673i)T \) |
| 53 | \( 1 + (0.739 + 0.673i)T \) |
| 59 | \( 1 + (0.739 + 0.673i)T \) |
| 61 | \( 1 + (0.445 + 0.895i)T \) |
| 67 | \( 1 + (-0.982 - 0.183i)T \) |
| 71 | \( 1 + (-0.982 - 0.183i)T \) |
| 73 | \( 1 + (0.445 + 0.895i)T \) |
| 79 | \( 1 + (-0.982 + 0.183i)T \) |
| 83 | \( 1 + (0.445 + 0.895i)T \) |
| 89 | \( 1 + (0.932 + 0.361i)T \) |
| 97 | \( 1 + (0.445 - 0.895i)T \) |
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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
−21.94053680556388626598749826520, −21.232246289655230812346467877322, −20.68127323464345004042331417571, −19.796843270849072706966407217863, −18.97739400157934601552786722400, −18.04368396306681444170415399104, −17.42887729235443596540648224445, −16.832209737533765087696602451790, −15.83357387720452912021821819094, −14.70061816801487348185576092836, −14.384065675962183104106149530825, −13.17515346560489621884031639958, −12.05253382903701615946645294088, −11.64672939781827515469637814928, −10.48786574844408457850977419835, −9.8437819758624302096059865277, −9.425205666377574073095028176378, −8.55701553590593065362697373185, −7.56873194073705603121521457190, −6.32770487410386248518095114265, −5.15458167250947455371828819288, −4.50989845909581797574512070255, −3.271794586426353229855801637404, −2.38448240175927228047875011525, −1.75561462745232627612843624841,
0.51695583733662540274911934784, 1.478458617983642592317102255523, 2.29826488438483356555421305955, 3.97724254517984400835296152535, 5.28233069457413674582422706294, 5.89140439624830512542673163978, 6.80659126803948013399217365037, 7.44846800561246190124692219502, 8.50222019990149923308479062583, 8.860493317003830526775946937884, 10.22957646232161611028769408198, 10.63283860255330099856720230444, 11.98314899443507437056151315387, 12.91521103498039077817042264766, 13.797000055545471556719204235410, 14.26429374424265302284788888816, 14.88203298369475561445364886074, 16.507438200771431310620213063598, 16.8745208354880491002524520408, 17.48970981326543466133581186303, 18.142920495549567690995324762544, 19.06872873689775378340703810160, 19.72191879408544228870678251054, 20.40654766672797963470371204250, 21.560642132708524636737989521734