L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.577 + 1.63i)3-s − 1.00i·4-s + (2.23 − 0.0135i)5-s + (−1.56 − 0.746i)6-s + (3.38 + 3.38i)7-s + (0.707 + 0.707i)8-s + (−2.33 + 1.88i)9-s + (−1.57 + 1.59i)10-s − 1.22i·11-s + (1.63 − 0.577i)12-s + (3.51 − 3.51i)13-s − 4.78·14-s + (1.31 + 3.64i)15-s − 1.00·16-s + (0.826 − 0.826i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.333 + 0.942i)3-s − 0.500i·4-s + (0.999 − 0.00606i)5-s + (−0.638 − 0.304i)6-s + (1.27 + 1.27i)7-s + (0.250 + 0.250i)8-s + (−0.777 + 0.628i)9-s + (−0.496 + 0.503i)10-s − 0.369i·11-s + (0.471 − 0.166i)12-s + (0.975 − 0.975i)13-s − 1.27·14-s + (0.339 + 0.940i)15-s − 0.250·16-s + (0.200 − 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12839 + 1.26526i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12839 + 1.26526i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.577 - 1.63i)T \) |
| 5 | \( 1 + (-2.23 + 0.0135i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (-3.38 - 3.38i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.22iT - 11T^{2} \) |
| 13 | \( 1 + (-3.51 + 3.51i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.826 + 0.826i)T - 17iT^{2} \) |
| 23 | \( 1 + (1.04 + 1.04i)T + 23iT^{2} \) |
| 29 | \( 1 + 8.14T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + (5.83 + 5.83i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.53iT - 41T^{2} \) |
| 43 | \( 1 + (2.68 - 2.68i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.84 - 6.84i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.40 + 3.40i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.91T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + (-6.00 - 6.00i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.35iT - 71T^{2} \) |
| 73 | \( 1 + (3.99 - 3.99i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.75iT - 79T^{2} \) |
| 83 | \( 1 + (-0.717 - 0.717i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.384T + 89T^{2} \) |
| 97 | \( 1 + (12.5 + 12.5i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.86226333910404088901496194809, −9.928116156162677416044370839593, −9.051633540673283752172351643785, −8.543530388902812518774266625052, −7.81187381356449460853688039455, −6.05817665444561266599212575712, −5.55628434775201960306391044696, −4.75829729418449954361275559765, −3.02720202039811421703698220135, −1.79638523556042124547928492351,
1.38237209253487188079518830034, 1.84045039813779506590546768091, 3.51403742489743781115176641624, 4.76448565861648576708628867687, 6.21467182508335585111279028828, 7.06961721137720124867455913456, 7.938461159903597647245528614770, 8.667763685134642970680802564538, 9.638947303384769098579610192052, 10.55062135694944555391581438817