L(s) = 1 | + (0.809 + 1.40i)2-s + (−0.309 + 0.535i)4-s − 5-s + (0.118 − 0.204i)7-s + 2.23·8-s + (−0.809 − 1.40i)10-s + (2.11 + 3.66i)11-s + (−1 + 3.46i)13-s + 0.381·14-s + (2.42 + 4.20i)16-s + (2.73 − 4.73i)17-s + (0.118 − 0.204i)19-s + (0.309 − 0.535i)20-s + (−3.42 + 5.93i)22-s + (4.11 + 7.13i)23-s + ⋯ |
L(s) = 1 | + (0.572 + 0.990i)2-s + (−0.154 + 0.267i)4-s − 0.447·5-s + (0.0446 − 0.0772i)7-s + 0.790·8-s + (−0.255 − 0.443i)10-s + (0.638 + 1.10i)11-s + (−0.277 + 0.960i)13-s + 0.102·14-s + (0.606 + 1.05i)16-s + (0.663 − 1.14i)17-s + (0.0270 − 0.0469i)19-s + (0.0690 − 0.119i)20-s + (−0.730 + 1.26i)22-s + (0.858 + 1.48i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46689 + 1.44820i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46689 + 1.44820i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 2 | \( 1 + (-0.809 - 1.40i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.118 + 0.204i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.11 - 3.66i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.73 + 4.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.118 + 0.204i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.11 - 7.13i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.736 - 1.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.97 + 5.14i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.881 + 1.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-6.35 + 11.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.20 + 10.7i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.35 + 9.27i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.881 - 1.52i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.73 - 4.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.09359413037000983723834205366, −9.836312274302520445828326044831, −9.205639202231114281460028411887, −7.83472348785181759999981285662, −7.15063051886026830451418508342, −6.63059812629052920961928605121, −5.26346336850704819485088613389, −4.66392160416161222538286199813, −3.54928074039369604531821340960, −1.68354101681500981589729540238,
1.14033081672855307634296520767, 2.80576361194186235420948973634, 3.55235295080124761867939471800, 4.55724722911197847853616482595, 5.68954410576967202273034888275, 6.83019166118608350711647652954, 8.055674514605799706124374871978, 8.582208120235336332416112633836, 10.05719486182582429758167488158, 10.64271676168755328733340301434