L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (2.22 + 0.176i)5-s − 0.999·6-s + (−3.02 + 1.74i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.84 − 1.26i)10-s + 5.35·11-s + (0.866 + 0.499i)12-s + (−2.41 + 1.39i)13-s + 3.48·14-s + (2.01 − 0.961i)15-s + (−0.5 + 0.866i)16-s + (0.818 + 0.472i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.996 + 0.0789i)5-s − 0.408·6-s + (−1.14 + 0.659i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.582 − 0.400i)10-s + 1.61·11-s + (0.249 + 0.144i)12-s + (−0.669 + 0.386i)13-s + 0.932·14-s + (0.521 − 0.248i)15-s + (−0.125 + 0.216i)16-s + (0.198 + 0.114i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.636017908\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.636017908\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-2.22 - 0.176i)T \) |
| 37 | \( 1 + (-6.05 - 0.576i)T \) |
good | 7 | \( 1 + (3.02 - 1.74i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 13 | \( 1 + (2.41 - 1.39i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.818 - 0.472i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.113 + 0.195i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 0.422iT - 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 - 9.36T + 31T^{2} \) |
| 41 | \( 1 + (-4.96 - 8.60i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 0.646iT - 43T^{2} \) |
| 47 | \( 1 + 2.57iT - 47T^{2} \) |
| 53 | \( 1 + (1.84 + 1.06i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.21 - 12.4i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.10 - 1.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.10 + 3.52i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.0551 + 0.0955i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 3.85iT - 73T^{2} \) |
| 79 | \( 1 + (8.57 + 14.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.732 - 0.423i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.12 + 8.88i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7.24iT - 97T^{2} \) |
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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
−9.514724128957773274572706376165, −9.342562953591073159857374971383, −8.517806537829064897985963687898, −7.32703388176403252852033656374, −6.43003971836038776276328896565, −6.06543214757365948303757190962, −4.43484104286495820944235922259, −3.18384311145511464870418049950, −2.41855119193194626430412471949, −1.27095820013497695990362227468,
0.983974880281472588684611079262, 2.40706700639910291063818693208, 3.53140167511938695535394750318, 4.63491614021749932575502782938, 5.91936616039031478029774870556, 6.56280304876975703033076310971, 7.27575782278625609710599856278, 8.360280400836957717628755435950, 9.366006516092709845142039782341, 9.599623120693159130991253969646