L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 4.35·7-s + 0.999·8-s + (−0.499 + 0.866i)10-s + 3·11-s + (2 − 3.46i)13-s + (2.17 + 3.77i)14-s + (−0.5 − 0.866i)16-s + (1.67 + 2.90i)17-s + (−2.17 − 3.77i)19-s + 0.999·20-s + (−1.5 − 2.59i)22-s + (−1.5 + 2.59i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 1.64·7-s + 0.353·8-s + (−0.158 + 0.273i)10-s + 0.904·11-s + (0.554 − 0.960i)13-s + (0.582 + 1.00i)14-s + (−0.125 − 0.216i)16-s + (0.407 + 0.705i)17-s + (−0.499 − 0.866i)19-s + 0.223·20-s + (−0.319 − 0.553i)22-s + (−0.312 + 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08972151185\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08972151185\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (2.17 + 3.77i)T \) |
good | 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.67 - 2.90i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.67 + 8.10i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (3.17 + 5.50i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.53 - 11.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.35 - 11.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.67 - 4.64i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.67 - 9.83i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.03 + 8.72i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.32 + 4.01i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.35 - 4.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-0.179 + 0.310i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.03 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.389455411995299776777622970716, −9.103262247315565055315551578130, −8.158703009977110020914698388258, −7.27264081228987235450930221886, −6.31450229182884193070222901673, −5.67677077081498020827974569314, −4.22039929493861696812019485317, −3.58760524932616782934775598887, −2.74312259110631103111163624658, −1.24266408656899429154458162155,
0.04139913275610040773446058515, 1.76146777944016288096384181229, 3.36437847291869407429132222012, 3.83449124895616672396621680171, 5.17705054317770685164448854091, 6.25457807235818676073625307857, 6.70391583270848617948473079400, 7.20588191637771834935359061899, 8.470738125672796555896624198072, 9.054777566115183596056381185707