L(s) = 1 | + (−0.181 − 0.983i)2-s + (−0.976 + 0.217i)3-s + (−0.934 + 0.357i)4-s + (0.744 + 0.667i)5-s + (0.391 + 0.920i)6-s + (−0.997 + 0.0729i)7-s + (0.520 + 0.853i)8-s + (0.905 − 0.424i)9-s + (0.520 − 0.853i)10-s + (−0.872 − 0.489i)11-s + (0.833 − 0.551i)12-s + (−0.181 − 0.983i)13-s + (0.252 + 0.967i)14-s + (−0.872 − 0.489i)15-s + (0.744 − 0.667i)16-s + (−0.694 − 0.719i)17-s + ⋯ |
L(s) = 1 | + (−0.181 − 0.983i)2-s + (−0.976 + 0.217i)3-s + (−0.934 + 0.357i)4-s + (0.744 + 0.667i)5-s + (0.391 + 0.920i)6-s + (−0.997 + 0.0729i)7-s + (0.520 + 0.853i)8-s + (0.905 − 0.424i)9-s + (0.520 − 0.853i)10-s + (−0.872 − 0.489i)11-s + (0.833 − 0.551i)12-s + (−0.181 − 0.983i)13-s + (0.252 + 0.967i)14-s + (−0.872 − 0.489i)15-s + (0.744 − 0.667i)16-s + (−0.694 − 0.719i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03617780510 - 0.3156947620i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03617780510 - 0.3156947620i\) |
\(L(1)\) |
\(\approx\) |
\(0.4635312896 - 0.2481272147i\) |
\(L(1)\) |
\(\approx\) |
\(0.4635312896 - 0.2481272147i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 173 | \( 1 \) |
good | 2 | \( 1 + (-0.181 - 0.983i)T \) |
| 3 | \( 1 + (-0.976 + 0.217i)T \) |
| 5 | \( 1 + (0.744 + 0.667i)T \) |
| 7 | \( 1 + (-0.997 + 0.0729i)T \) |
| 11 | \( 1 + (-0.872 - 0.489i)T \) |
| 13 | \( 1 + (-0.181 - 0.983i)T \) |
| 17 | \( 1 + (-0.694 - 0.719i)T \) |
| 19 | \( 1 + (0.252 - 0.967i)T \) |
| 23 | \( 1 + (-0.872 + 0.489i)T \) |
| 29 | \( 1 + (0.391 - 0.920i)T \) |
| 31 | \( 1 + (-0.976 - 0.217i)T \) |
| 37 | \( 1 + (0.252 - 0.967i)T \) |
| 41 | \( 1 + (-0.997 + 0.0729i)T \) |
| 43 | \( 1 + (-0.934 - 0.357i)T \) |
| 47 | \( 1 + (0.639 + 0.768i)T \) |
| 53 | \( 1 + (-0.0365 - 0.999i)T \) |
| 59 | \( 1 + (0.989 - 0.145i)T \) |
| 61 | \( 1 + (-0.694 + 0.719i)T \) |
| 67 | \( 1 + (-0.976 + 0.217i)T \) |
| 71 | \( 1 + (0.391 - 0.920i)T \) |
| 73 | \( 1 + (-0.457 - 0.889i)T \) |
| 79 | \( 1 + (0.639 - 0.768i)T \) |
| 83 | \( 1 + (0.109 - 0.994i)T \) |
| 89 | \( 1 + (-0.791 + 0.611i)T \) |
| 97 | \( 1 + (0.109 + 0.994i)T \) |
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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
−28.15160451200204634954155395434, −26.81379640825753297280185669200, −25.87255312979836138182577416146, −25.05255785149441798665853568657, −23.944876842018172062022332488489, −23.47453471340632750154553492605, −22.25009859221766705765462112125, −21.65110239696573789917984209842, −20.04753422157528440484282429280, −18.64622047700652408707162628246, −18.04649667713937778617107354236, −16.81585343078321997887992881825, −16.49533381941809517432078405971, −15.479916037897156272548039423160, −13.93122808397778137225646846483, −12.99701575784803191476763496477, −12.321648671751350077793143932749, −10.369876753624201895139897880346, −9.76555914898567677072743814182, −8.48396796226020046847338804144, −7.03291803686734924697927765895, −6.231968657684267026108098611001, −5.30995416891750751109932426659, −4.24094911842429801465021336887, −1.68117382398646057211054826455,
0.30328110664838275597465572220, 2.38034542800217392448969466394, 3.44451520832727331938863337480, 5.08986250776203504345501577439, 6.01778042018374017843333344647, 7.40144158300704450484620533129, 9.2322696065685842740914773340, 10.093481771094467701010853359451, 10.75531241017944639420526180477, 11.77794808614808192704008693976, 13.065176948360495846296297527986, 13.530244701012157314846730249378, 15.32501006074539006718099909307, 16.38460097552909135496990793192, 17.70451629151816423380556842673, 18.082941983269264228155137181617, 19.131942968176594414419012412072, 20.32234919022946931048045123927, 21.49179948370690401811488025807, 22.191110626477776052898688076908, 22.69087436851044776447717277759, 23.80581036440874227438901397175, 25.39526096011286648975846965283, 26.43707994148127920261413880879, 27.07725706698189577663620100975