L(s) = 1 | + (−0.606 + 0.795i)2-s + (−0.973 + 0.227i)3-s + (−0.264 − 0.964i)4-s + (−0.953 − 0.301i)5-s + (0.409 − 0.912i)6-s + (0.477 − 0.878i)7-s + (0.927 + 0.373i)8-s + (0.896 − 0.443i)9-s + (0.817 − 0.575i)10-s + (−0.997 + 0.0765i)11-s + (0.477 + 0.878i)12-s + (0.114 + 0.993i)13-s + (0.409 + 0.912i)14-s + (0.997 + 0.0765i)15-s + (−0.859 + 0.511i)16-s + (0.338 − 0.941i)17-s + ⋯ |
L(s) = 1 | + (−0.606 + 0.795i)2-s + (−0.973 + 0.227i)3-s + (−0.264 − 0.964i)4-s + (−0.953 − 0.301i)5-s + (0.409 − 0.912i)6-s + (0.477 − 0.878i)7-s + (0.927 + 0.373i)8-s + (0.896 − 0.443i)9-s + (0.817 − 0.575i)10-s + (−0.997 + 0.0765i)11-s + (0.477 + 0.878i)12-s + (0.114 + 0.993i)13-s + (0.409 + 0.912i)14-s + (0.997 + 0.0765i)15-s + (−0.859 + 0.511i)16-s + (0.338 − 0.941i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2696014964 + 0.3724207740i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2696014964 + 0.3724207740i\) |
\(L(1)\) |
\(\approx\) |
\(0.4512398175 + 0.1642167928i\) |
\(L(1)\) |
\(\approx\) |
\(0.4512398175 + 0.1642167928i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (-0.606 + 0.795i)T \) |
| 3 | \( 1 + (-0.973 + 0.227i)T \) |
| 5 | \( 1 + (-0.953 - 0.301i)T \) |
| 7 | \( 1 + (0.477 - 0.878i)T \) |
| 11 | \( 1 + (-0.997 + 0.0765i)T \) |
| 13 | \( 1 + (0.114 + 0.993i)T \) |
| 17 | \( 1 + (0.338 - 0.941i)T \) |
| 19 | \( 1 + (-0.720 - 0.693i)T \) |
| 23 | \( 1 + (0.190 + 0.981i)T \) |
| 29 | \( 1 + (0.0383 + 0.999i)T \) |
| 31 | \( 1 + (-0.927 + 0.373i)T \) |
| 37 | \( 1 + (0.896 + 0.443i)T \) |
| 41 | \( 1 + (0.606 + 0.795i)T \) |
| 43 | \( 1 + (0.771 + 0.636i)T \) |
| 47 | \( 1 + (0.665 + 0.746i)T \) |
| 53 | \( 1 + (0.665 - 0.746i)T \) |
| 59 | \( 1 + (0.988 + 0.152i)T \) |
| 61 | \( 1 + (-0.543 + 0.839i)T \) |
| 67 | \( 1 + (0.859 - 0.511i)T \) |
| 71 | \( 1 + (-0.477 - 0.878i)T \) |
| 73 | \( 1 + (-0.817 + 0.575i)T \) |
| 79 | \( 1 + (0.264 + 0.964i)T \) |
| 89 | \( 1 + (0.409 - 0.912i)T \) |
| 97 | \( 1 + (0.409 + 0.912i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.25576069853478239275502130759, −29.07223876391406045145628809237, −28.038763396921737156029455086112, −27.58441598886798868987023447269, −26.43928137990570164119491422313, −25.00040357214370225868850897381, −23.62410972240772026797741719919, −22.68146745547304185702929886528, −21.64924736297250727811017383270, −20.5784953446243571999503693023, −19.003330603286973691522100174229, −18.52653459235147582386785206803, −17.45866944712261144357897145652, −16.1998447101818935177082053027, −15.09211355197617951370521815266, −12.82964052746539480673546943295, −12.23887926377529740287001650649, −11.020424631556517463526441265924, −10.39529628578485058374536384373, −8.390542974765176664705039539372, −7.58893814115842943170493301980, −5.70984355545620612387710236685, −4.12928182101539646902076064978, −2.35790393130318285949293550039, −0.402739836777993387404351904800,
0.98920508162690811190067625779, 4.31887983926217837028762641186, 5.19543397518839733313180293285, 6.92396554467860660421593532895, 7.70068898830542813802283493725, 9.27121455009531620090497131797, 10.70910448235224851590559815727, 11.44908900592514012375589329743, 13.16346809086069457176287896079, 14.72436624283839836591775094933, 15.981908986697002835799139664218, 16.5049973200990274537552953107, 17.66443625589300689638343098188, 18.65136919837688583660757840349, 19.91787258154667257586234901910, 21.22182556198270802018188767409, 22.92587812785220795475206637028, 23.71039441326847195919142669902, 24.02468960453349683907760333738, 25.84702643984628284141968951356, 26.96526502924026533720648851016, 27.50307889998755131483915038484, 28.53838684782812940712681795909, 29.50025306187977221590119142911, 31.13705008712019023541060765915