L(s) = 1 | + (1 + 1.73i)4-s + 5-s + (0.5 + 0.866i)7-s + (3 − 5.19i)11-s + (2.5 − 2.59i)13-s + (−1.99 + 3.46i)16-s + (2 + 3.46i)19-s + (1 + 1.73i)20-s + (−3 + 5.19i)23-s + 25-s + (−0.999 + 1.73i)28-s + (−3 + 5.19i)29-s + 5·31-s + (0.5 + 0.866i)35-s + (−1 + 1.73i)37-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)4-s + 0.447·5-s + (0.188 + 0.327i)7-s + (0.904 − 1.56i)11-s + (0.693 − 0.720i)13-s + (−0.499 + 0.866i)16-s + (0.458 + 0.794i)19-s + (0.223 + 0.387i)20-s + (−0.625 + 1.08i)23-s + 0.200·25-s + (−0.188 + 0.327i)28-s + (−0.557 + 0.964i)29-s + 0.898·31-s + (0.0845 + 0.146i)35-s + (−0.164 + 0.284i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.83487 + 0.504284i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.83487 + 0.504284i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.5 + 2.59i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.95939909869429806902724756195, −9.940166579244293426274106700457, −8.684852461348152437677356590353, −8.368067528462360192854415895141, −7.21925409398443950937354935443, −6.15321688625486100375517154056, −5.52156313861216815334220146915, −3.75458208441060666867643746289, −3.14382143384451334832297267850, −1.55100035165527543656502168110,
1.35605156967821264572409554034, 2.35690980881093919083696393558, 4.18638496210616983215843022608, 4.99511264854779931031422214952, 6.40232625839597325705105545264, 6.67584433257157179003630504074, 7.85558090765350425233959834146, 9.252614083169208074026220082783, 9.702889871145250923174242109704, 10.56260755059863972692153469529