L(s) = 1 | + (−0.5 − 0.866i)2-s + (1 − 1.73i)3-s + (0.500 − 0.866i)4-s + (−1 − 1.73i)5-s − 1.99·6-s − 3·8-s + (−0.499 − 0.866i)9-s + (−0.999 + 1.73i)10-s + (−0.5 + 0.866i)11-s + (−0.999 − 1.73i)12-s − 4·13-s − 3.99·15-s + (0.500 + 0.866i)16-s + (2 − 3.46i)17-s + (−0.499 + 0.866i)18-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.577 − 0.999i)3-s + (0.250 − 0.433i)4-s + (−0.447 − 0.774i)5-s − 0.816·6-s − 1.06·8-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (−0.150 + 0.261i)11-s + (−0.288 − 0.499i)12-s − 1.10·13-s − 1.03·15-s + (0.125 + 0.216i)16-s + (0.485 − 0.840i)17-s + (−0.117 + 0.204i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.148273 + 1.16353i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.148273 + 1.16353i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (4 - 6.92i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−10.16352196279216803423384725093, −9.531001684614036680414375611918, −8.604315995159997365852291002654, −7.65390531723111661333422869106, −7.04171009416625584697534537626, −5.69366575655268010007718462092, −4.62513542852633258320781217837, −2.92424844745527551945513742561, −1.98172644318346283194013118055, −0.69879789330897275188748878624,
2.75367979119873177476454494544, 3.43296212852972034518347041345, 4.53986068225512843222698464083, 5.95800371243950598455606320016, 7.05166387642353150669127418995, 7.69088727673016874897390417370, 8.654573274737109421972609295479, 9.360059878873926623215291788329, 10.35417603145711302199356990572, 11.01399970901142558658085411459