L(s) = 1 | + (0.739 − 0.673i)2-s + (0.932 − 0.361i)3-s + (0.0922 − 0.995i)4-s + (−0.850 − 0.526i)5-s + (0.445 − 0.895i)6-s + (−0.602 + 0.798i)7-s + (−0.602 − 0.798i)8-s + (0.739 − 0.673i)9-s + (−0.982 + 0.183i)10-s + (0.739 − 0.673i)11-s + (−0.273 − 0.961i)12-s + (−0.602 + 0.798i)13-s + (0.0922 + 0.995i)14-s + (−0.982 − 0.183i)15-s + (−0.982 − 0.183i)16-s + (0.445 + 0.895i)17-s + ⋯ |
L(s) = 1 | + (0.739 − 0.673i)2-s + (0.932 − 0.361i)3-s + (0.0922 − 0.995i)4-s + (−0.850 − 0.526i)5-s + (0.445 − 0.895i)6-s + (−0.602 + 0.798i)7-s + (−0.602 − 0.798i)8-s + (0.739 − 0.673i)9-s + (−0.982 + 0.183i)10-s + (0.739 − 0.673i)11-s + (−0.273 − 0.961i)12-s + (−0.602 + 0.798i)13-s + (0.0922 + 0.995i)14-s + (−0.982 − 0.183i)15-s + (−0.982 − 0.183i)16-s + (0.445 + 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0419 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0419 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.140802978 - 1.189696852i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.140802978 - 1.189696852i\) |
\(L(1)\) |
\(\approx\) |
\(1.360066744 - 0.8717924032i\) |
\(L(1)\) |
\(\approx\) |
\(1.360066744 - 0.8717924032i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 103 | \( 1 \) |
good | 2 | \( 1 + (0.739 - 0.673i)T \) |
| 3 | \( 1 + (0.932 - 0.361i)T \) |
| 5 | \( 1 + (-0.850 - 0.526i)T \) |
| 7 | \( 1 + (-0.602 + 0.798i)T \) |
| 11 | \( 1 + (0.739 - 0.673i)T \) |
| 13 | \( 1 + (-0.602 + 0.798i)T \) |
| 17 | \( 1 + (0.445 + 0.895i)T \) |
| 19 | \( 1 + (0.932 + 0.361i)T \) |
| 23 | \( 1 + (0.739 + 0.673i)T \) |
| 29 | \( 1 + (-0.850 - 0.526i)T \) |
| 31 | \( 1 + (-0.982 - 0.183i)T \) |
| 37 | \( 1 + (-0.273 - 0.961i)T \) |
| 41 | \( 1 + (-0.850 + 0.526i)T \) |
| 43 | \( 1 + (-0.273 + 0.961i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.932 + 0.361i)T \) |
| 59 | \( 1 + (-0.602 - 0.798i)T \) |
| 61 | \( 1 + (0.445 + 0.895i)T \) |
| 67 | \( 1 + (-0.602 - 0.798i)T \) |
| 71 | \( 1 + (-0.850 + 0.526i)T \) |
| 73 | \( 1 + (-0.850 + 0.526i)T \) |
| 79 | \( 1 + (-0.850 - 0.526i)T \) |
| 83 | \( 1 + (-0.602 + 0.798i)T \) |
| 89 | \( 1 + (0.0922 + 0.995i)T \) |
| 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
−30.454894206285918156590113937204, −29.55595424066230905324625910838, −27.3861461862216298579237236505, −26.85397847527864224451272511079, −25.83478013403104345443589375524, −25.0336274432218032866396111860, −23.86771445931088540899334772527, −22.59480542376526760296506364634, −22.249855559550315676669003838348, −20.38146738645526125089310009212, −20.0716426711943265052176054935, −18.59310572458332865517451389436, −16.95872957774105138356258634806, −15.93812753222040188143229578920, −15.0096166668954899316632027329, −14.26716680710627832940176314343, −13.134200907261336809116952677094, −11.93894859249209497725872080911, −10.34820917193139078478897277420, −8.97111980141245340849749766808, −7.35486829946460677828287205912, −7.16296208881469373879780420517, −4.93060930895517019847512958055, −3.72761094299364522468608076907, −2.94104912582558894217159137118,
1.55657860076979911948408245626, 3.16370987827969657185360309463, 4.0144558378376591887840390920, 5.72114768818212864072680181299, 7.22612539074699924418348833165, 8.82721984011645036442618390280, 9.58960303429168735579443538631, 11.521683432473167706630589486293, 12.31018613914307323330603560411, 13.228907141936707428645636775593, 14.48121101762412024271083626350, 15.290355633516756591477428054497, 16.47879327379581488143909477864, 18.6929150957765598343466984170, 19.283682303748319716398990249256, 19.961695452905480351987654936506, 21.17723128191632566881864854038, 22.06531021986202076646995927138, 23.4073568375505289719573322447, 24.36472021496212213056940215748, 25.004203741445844838310630475415, 26.55746599709175935224825864841, 27.63624660847634166558859630093, 28.73017992733886613433517064063, 29.70100978530618981190354672932