L(s) = 1 | + (0.682 + 0.730i)2-s + (−0.990 + 0.136i)3-s + (−0.0682 + 0.997i)4-s + (−0.460 − 0.887i)5-s + (−0.775 − 0.631i)6-s + (−0.917 − 0.398i)7-s + (−0.775 + 0.631i)8-s + (0.962 − 0.269i)9-s + (0.334 − 0.942i)10-s + (−0.203 − 0.979i)11-s + (−0.0682 − 0.997i)12-s + (−0.854 + 0.519i)13-s + (−0.334 − 0.942i)14-s + (0.576 + 0.816i)15-s + (−0.990 − 0.136i)16-s + (0.203 − 0.979i)17-s + ⋯ |
L(s) = 1 | + (0.682 + 0.730i)2-s + (−0.990 + 0.136i)3-s + (−0.0682 + 0.997i)4-s + (−0.460 − 0.887i)5-s + (−0.775 − 0.631i)6-s + (−0.917 − 0.398i)7-s + (−0.775 + 0.631i)8-s + (0.962 − 0.269i)9-s + (0.334 − 0.942i)10-s + (−0.203 − 0.979i)11-s + (−0.0682 − 0.997i)12-s + (−0.854 + 0.519i)13-s + (−0.334 − 0.942i)14-s + (0.576 + 0.816i)15-s + (−0.990 − 0.136i)16-s + (0.203 − 0.979i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09090031094 - 0.1520642683i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09090031094 - 0.1520642683i\) |
\(L(1)\) |
\(\approx\) |
\(0.6483900853 + 0.1637882619i\) |
\(L(1)\) |
\(\approx\) |
\(0.6483900853 + 0.1637882619i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 47 | \( 1 \) |
good | 2 | \( 1 + (0.682 + 0.730i)T \) |
| 3 | \( 1 + (-0.990 + 0.136i)T \) |
| 5 | \( 1 + (-0.460 - 0.887i)T \) |
| 7 | \( 1 + (-0.917 - 0.398i)T \) |
| 11 | \( 1 + (-0.203 - 0.979i)T \) |
| 13 | \( 1 + (-0.854 + 0.519i)T \) |
| 17 | \( 1 + (0.203 - 0.979i)T \) |
| 19 | \( 1 + (-0.460 + 0.887i)T \) |
| 23 | \( 1 + (-0.682 + 0.730i)T \) |
| 29 | \( 1 + (-0.854 - 0.519i)T \) |
| 31 | \( 1 + (0.990 + 0.136i)T \) |
| 37 | \( 1 + (-0.334 + 0.942i)T \) |
| 41 | \( 1 + (0.775 + 0.631i)T \) |
| 43 | \( 1 + (0.0682 - 0.997i)T \) |
| 53 | \( 1 + (-0.775 - 0.631i)T \) |
| 59 | \( 1 + (-0.0682 - 0.997i)T \) |
| 61 | \( 1 + (-0.334 - 0.942i)T \) |
| 67 | \( 1 + (0.917 - 0.398i)T \) |
| 71 | \( 1 + (0.682 - 0.730i)T \) |
| 73 | \( 1 + (-0.962 - 0.269i)T \) |
| 79 | \( 1 + (-0.576 - 0.816i)T \) |
| 83 | \( 1 + (0.203 + 0.979i)T \) |
| 89 | \( 1 + (0.460 + 0.887i)T \) |
| 97 | \( 1 + (-0.990 + 0.136i)T \) |
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
−34.14621164426757029617279257021, −32.95624215456207344926660660319, −31.77792552441328241974070714157, −30.41979394702782578727956683195, −29.739450338503376248794104810553, −28.48354629276440470597874853153, −27.76638773782971598294415250732, −26.13528042846900489673922835988, −24.37010926002189665642852721394, −23.113237800512321266535604753188, −22.49488303343360710616426698876, −21.648949247579389937750618331706, −19.78313357462792453099583052279, −18.86289512897531407120589094649, −17.65945267572827467497284750106, −15.74922962635524499411981594971, −14.811634787833235320145240005771, −12.86071788522796846114162075013, −12.17662127085474880749385357371, −10.76827732433700900320767425882, −9.89527878290563771103510178979, −7.06723543688232798632296661106, −5.89583269931243232047475303230, −4.27274887575579122360871816428, −2.49889865474376245639814594794,
0.09037887307542898642511897305, 3.7595494339243556303102447294, 5.06963202173982454217726326472, 6.32195389624856860725825111047, 7.7501584457690818506984709563, 9.57692696491010360657690204468, 11.59827593835403683414897967769, 12.5351588726623900661589113609, 13.71861324706424442490631327846, 15.6639971545157199079290665755, 16.44308588265338272635670759821, 17.13473187654977259904745040162, 18.98433718238383381932305882545, 20.747427800553631529879976693867, 21.94514118530183719135093982951, 23.019582079423965081922287429292, 23.87920387755040108636537834435, 24.82124243812283740447406132892, 26.49960931533020101273658372360, 27.453754576063209954295303791384, 29.04672391964313977806618455716, 29.762173025006334310619932498906, 31.75957366673770050612613846508, 32.13295946646907090865616287656, 33.47535230349158465643256313684