L(s) = 1 | + (−0.999 + 0.0361i)2-s + (−0.773 − 0.633i)3-s + (0.997 − 0.0721i)4-s + (0.958 + 0.284i)5-s + (0.796 + 0.605i)6-s + (−0.0901 − 0.995i)7-s + (−0.994 + 0.108i)8-s + (0.197 + 0.980i)9-s + (−0.968 − 0.250i)10-s + (−0.370 + 0.928i)11-s + (−0.817 − 0.576i)12-s + (0.976 − 0.214i)13-s + (0.126 + 0.992i)14-s + (−0.561 − 0.827i)15-s + (0.989 − 0.143i)16-s + (−0.856 + 0.515i)17-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0361i)2-s + (−0.773 − 0.633i)3-s + (0.997 − 0.0721i)4-s + (0.958 + 0.284i)5-s + (0.796 + 0.605i)6-s + (−0.0901 − 0.995i)7-s + (−0.994 + 0.108i)8-s + (0.197 + 0.980i)9-s + (−0.968 − 0.250i)10-s + (−0.370 + 0.928i)11-s + (−0.817 − 0.576i)12-s + (0.976 − 0.214i)13-s + (0.126 + 0.992i)14-s + (−0.561 − 0.827i)15-s + (0.989 − 0.143i)16-s + (−0.856 + 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.114312962 - 0.1194534686i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.114312962 - 0.1194534686i\) |
\(L(1)\) |
\(\approx\) |
\(0.7238399382 - 0.09085904054i\) |
\(L(1)\) |
\(\approx\) |
\(0.7238399382 - 0.09085904054i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4003 | \( 1 \) |
good | 2 | \( 1 + (-0.999 + 0.0361i)T \) |
| 3 | \( 1 + (-0.773 - 0.633i)T \) |
| 5 | \( 1 + (0.958 + 0.284i)T \) |
| 7 | \( 1 + (-0.0901 - 0.995i)T \) |
| 11 | \( 1 + (-0.370 + 0.928i)T \) |
| 13 | \( 1 + (0.976 - 0.214i)T \) |
| 17 | \( 1 + (-0.856 + 0.515i)T \) |
| 19 | \( 1 + (0.874 + 0.484i)T \) |
| 23 | \( 1 + (0.989 - 0.143i)T \) |
| 29 | \( 1 + (0.874 - 0.484i)T \) |
| 31 | \( 1 + (-0.436 + 0.899i)T \) |
| 37 | \( 1 + (0.874 - 0.484i)T \) |
| 41 | \( 1 + (-0.370 - 0.928i)T \) |
| 43 | \( 1 + (0.935 + 0.353i)T \) |
| 47 | \( 1 + (0.989 + 0.143i)T \) |
| 53 | \( 1 + (0.989 + 0.143i)T \) |
| 59 | \( 1 + (0.403 - 0.915i)T \) |
| 61 | \( 1 + (-0.922 - 0.386i)T \) |
| 67 | \( 1 + (0.126 - 0.992i)T \) |
| 71 | \( 1 + (-0.773 - 0.633i)T \) |
| 73 | \( 1 + (0.0541 + 0.998i)T \) |
| 79 | \( 1 + (-0.922 + 0.386i)T \) |
| 83 | \( 1 + (-0.999 + 0.0361i)T \) |
| 89 | \( 1 + (0.126 + 0.992i)T \) |
| 97 | \( 1 + (0.126 - 0.992i)T \) |
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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
−18.38826510846693977897352541822, −17.87565646656288424440227499266, −17.22899240233845433175622656882, −16.3666855677802702609638544857, −16.11349626331559467715481646269, −15.4327369370222180161876274801, −14.73066297210466521418433861904, −13.5531161436476369650540771114, −13.01285434248849397690981285013, −11.98731501331256724132376649501, −11.4120961928185412185936893749, −10.910476861308505581426507878255, −10.17031530407360429175562626678, −9.36204999264475334428083271253, −8.94851103043276657574226603632, −8.504368544383578151664097089824, −7.18254732791500508617895690264, −6.42839429012839422383317348693, −5.772535459759040596286609499089, −5.431348439348898162368145149755, −4.372839935280752663306480665316, −3.02673256595087436764678326859, −2.65792427533935438633873770121, −1.39679611363130564742330836889, −0.70823040559955258801025753228,
0.8069417223069318767422462084, 1.420707616360035850176529776294, 2.16698227361361167218931696667, 3.037446341388832867686782006021, 4.257564799904615312747461939738, 5.31582769102519027197232990119, 5.98260208054695091933117185763, 6.721663045373170686599779869355, 7.138209810511734764354091451862, 7.83293540792611747501368787002, 8.73367578988924267657857904355, 9.596475780105094067793481241691, 10.325807959798970130343333523256, 10.75494645884189220065666056739, 11.210589522239170005737486427, 12.336068148203903504925376025870, 12.86897464172973449186174153623, 13.61616524598826676417444675953, 14.25355347055859101264977666127, 15.31058547036554077854083248696, 16.01100419160907533863051980791, 16.67706719469620631626818020490, 17.473398081530100244689859498860, 17.60453470742259557710174075564, 18.2992492162757718744727968886