L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 2·5-s + (0.499 + 0.866i)6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)10-s + (2 − 3.46i)11-s + 0.999·12-s + (−3.5 − 0.866i)13-s + 0.999·14-s + (1 − 1.73i)15-s + (−0.5 + 0.866i)16-s + (−3.5 − 6.06i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.894·5-s + (0.204 + 0.353i)6-s + (0.188 + 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (0.603 − 1.04i)11-s + 0.288·12-s + (−0.970 − 0.240i)13-s + 0.267·14-s + (0.258 − 0.447i)15-s + (−0.125 + 0.216i)16-s + (−0.848 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.184932 - 0.672890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.184932 - 0.672890i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (3.5 + 0.866i)T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 12T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + T + 83T^{2} \) |
| 89 | \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.80529507801255765824642357014, −9.511576942030541412130134295292, −8.978480425123472703572660406686, −7.81221791838822531039387653706, −6.69770790918759634308999076011, −5.47648572000703580374583960222, −4.62553111551699653649198192288, −3.67833373829573721776399603891, −2.55804641093132939198978389761, −0.36138729858545339694152960983,
1.95500335569753430586389179570, 3.85197468361206663828275367992, 4.48732756969274897181065046809, 5.71462156459475568842742310562, 6.88700727658138136191215866695, 7.33803885625386128591416180320, 8.218561257307482672033682800642, 9.199742222134130412445483437910, 10.41134461087545982118472918663, 11.34158752982643437939785428671