L(s) = 1 | + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)17-s + (−0.809 + 0.587i)19-s − 21-s − 23-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)35-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)17-s + (−0.809 + 0.587i)19-s − 21-s − 23-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5379672705 + 0.7222362654i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5379672705 + 0.7222362654i\) |
\(L(1)\) |
\(\approx\) |
\(0.7544201683 + 0.2254452602i\) |
\(L(1)\) |
\(\approx\) |
\(0.7544201683 + 0.2254452602i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.809 + 0.587i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.809 - 0.587i)T \) |
| 53 | \( 1 + (-0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.809 - 0.587i)T \) |
| 61 | \( 1 + (-0.309 - 0.951i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.157471266760955339019509104429, −29.206587494723186983995216007898, −27.70655267128661182189185200892, −27.16185103480194576044785034307, −25.76318759561328284788759482390, −24.51272350827212491144800694670, −23.51900252688907124016888526589, −22.73383990359546690981187964629, −21.75031347072144199337214722873, −20.23044707444550582907302132238, −19.02804251005549747228003278223, −17.9779632035732095276997816909, −17.31049473321381449245979693407, −15.85561384648880259102292210610, −14.555518108344083399394838360463, −13.457200432559701110977975652667, −12.01852821742871836922482544763, −11.073631809210209041022904154547, −10.20228397442539215374316570459, −7.93883490845675378442245332401, −7.188318316037918327520181603262, −5.84619498761786799930468828347, −4.375267203339177563519492033082, −2.42006020731366506099652799193, −0.48039704911832046141342113778,
1.54908070691974308815909262611, 4.07661181760113763867905255468, 5.00204012520231684109689979898, 6.21846396710851520121857741438, 8.12380395916241158045170991383, 9.20658259786213579189227322050, 10.60501969387131997435116337733, 11.88220812089565766902717285000, 12.47477288389112366776316117739, 14.38070898743556597492460844903, 15.5239456843187996220246169586, 16.57875295019115192410832754313, 17.37087588201107213450517135859, 18.64727375899107117811179121548, 20.09501667301295572888085489169, 21.31945331335786340483824991135, 21.77498525093982904737272076387, 23.516125664946322230304971862492, 23.92889141244392233077568412562, 25.24645847054428223901972314260, 26.7587561200385200324488019450, 27.7060162870340339058512820666, 28.32242011369709066519646591276, 29.27058989329998240294421071537, 30.749049288467832805811965833300