L(s) = 1 | + (0.142 − 0.989i)3-s + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.654 + 0.755i)13-s + (0.142 + 0.989i)15-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.654 + 0.755i)33-s + ⋯ |
L(s) = 1 | + (0.142 − 0.989i)3-s + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.654 + 0.755i)13-s + (0.142 + 0.989i)15-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.654 + 0.755i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3609339013 + 0.4707334798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3609339013 + 0.4707334798i\) |
\(L(1)\) |
\(\approx\) |
\(0.7654584523 + 0.009442246281i\) |
\(L(1)\) |
\(\approx\) |
\(0.7654584523 + 0.009442246281i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 11 | \( 1 + (-0.841 - 0.540i)T \) |
| 13 | \( 1 + (-0.654 + 0.755i)T \) |
| 17 | \( 1 + (0.415 + 0.909i)T \) |
| 19 | \( 1 + (-0.415 + 0.909i)T \) |
| 29 | \( 1 + (0.415 + 0.909i)T \) |
| 31 | \( 1 + (0.142 + 0.989i)T \) |
| 37 | \( 1 + (-0.959 - 0.281i)T \) |
| 41 | \( 1 + (-0.959 + 0.281i)T \) |
| 43 | \( 1 + (0.142 - 0.989i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.654 - 0.755i)T \) |
| 59 | \( 1 + (0.654 - 0.755i)T \) |
| 61 | \( 1 + (-0.142 - 0.989i)T \) |
| 67 | \( 1 + (-0.841 + 0.540i)T \) |
| 71 | \( 1 + (-0.841 + 0.540i)T \) |
| 73 | \( 1 + (0.415 - 0.909i)T \) |
| 79 | \( 1 + (0.654 - 0.755i)T \) |
| 83 | \( 1 + (0.959 + 0.281i)T \) |
| 89 | \( 1 + (-0.142 + 0.989i)T \) |
| 97 | \( 1 + (-0.959 + 0.281i)T \) |
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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
−29.90681644371988076110761214678, −28.41037895222802969396750367147, −27.53837454025864757848949012404, −26.8739642162069532490108583533, −25.89139786912172042344138917453, −24.4365642392819295717847253390, −23.30676434880302795289457371726, −22.52976024596552945675218687658, −20.99009681264095575212113345030, −20.394547154376053219516895406145, −19.43461678260867554181970081597, −17.743178415287845299040243054570, −16.69947036614961466434156567500, −15.550933665251795263249011125297, −14.877366698364558381825355043088, −13.45291211141360503767616409127, −11.900214757039425289997454785324, −10.837127487392737825638686859499, −9.81127882447923184281367410825, −8.27862305319517535542387475323, −7.41266034748705308573740773115, −5.09218855472529185117519465517, −4.38662591635681580779828266701, −2.88180005706009226167018485267, −0.25717312839695553691736600925,
1.82433457638488083222961275490, 3.29899585809516379239915982403, 5.18879428516421661926636101851, 6.665569798429187806374124009445, 7.94801088131005125513666019733, 8.59035756311435272419113939475, 10.69246598513412153888661645156, 11.89472486214394319899805281010, 12.55957908020031139731707733608, 14.173007515317698068376705419900, 14.97667585273362765150638654586, 16.358972487733017826125654373884, 17.77339558328210570413680342685, 18.86232345193947901994829125656, 19.302222185883445584007332867895, 20.77303233043927934422898637773, 21.97002788151570586704293931793, 23.48108409008183249659995753269, 23.900834409428376290238841041529, 25.00790679573916523042076059286, 26.19106547784125868185677283558, 27.25654418344104116402239870065, 28.42281370744912633195979789646, 29.42573934534099273200849215465, 30.60785419349033517523922898618