L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 + 0.984i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.766 − 0.642i)10-s + (−0.766 + 0.642i)11-s + 12-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)14-s + (0.939 − 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 + 0.984i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.766 − 0.642i)10-s + (−0.766 + 0.642i)11-s + 12-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)14-s + (0.939 − 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0991 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0991 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.459362189 - 1.321127551i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.459362189 - 1.321127551i\) |
\(L(1)\) |
\(\approx\) |
\(1.355531701 - 0.3525192406i\) |
\(L(1)\) |
\(\approx\) |
\(1.355531701 - 0.3525192406i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 7 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.766 + 0.642i)T \) |
| 13 | \( 1 + (-0.939 - 0.342i)T \) |
| 17 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.939 - 0.342i)T \) |
| 31 | \( 1 + (0.939 + 0.342i)T \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.173 - 0.984i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−18.506898717512472629364183097224, −17.93697211449503557353678328171, −17.23609566374046213422601530399, −16.585162524932775657071430747295, −15.89960420877324888029274765324, −14.91700097845929844353376552472, −14.40620473665126307141599357357, −13.89832236936869474609085481544, −13.43166410326460607255676148609, −12.63961664324924577774115749339, −11.82346135507668487602382711560, −11.35482413582725892540431816356, −10.53960242264980764229868692817, −9.6277787346558545515010917506, −8.45518219902394572708202681378, −7.705816542885121309016530445255, −7.48356654915663177134410935278, −6.69208907894530680693050397188, −6.22611469333496820822792347540, −5.21327341458918912486990146243, −4.52004368525953695387299309144, −3.34958060361079952061615781324, −2.975218818626683540902505354964, −2.22800376700741462398963955047, −0.910928683655202093021091572602,
0.45786274694666536721937995933, 1.80276244319798391695263476503, 2.56533752513031130230295069767, 3.1380277508302648520060093171, 4.229740823563581822145516799215, 4.83280672835771049555012186066, 5.20785820493083629585447951652, 5.81265113639397729513468260913, 6.98521023090204749831016672588, 8.13740082439469669681213104627, 8.74805270306771809568219534983, 9.44959286523883127232766928475, 10.076584884001292770867684191632, 10.70413657854445686857795548286, 11.57957607490111582428584177829, 12.23299305441815304840046465186, 12.65516203060632545664778614203, 13.46034769646992746034489466831, 14.24153254781572243392357982784, 15.18926600063940246811773457442, 15.42221945093719714210903986006, 15.78144346566446890560450155540, 17.04603524442055737764604219739, 17.38749047702707783073571986719, 18.49807210731212512070373238715