| L(s) = 1 | + (−1.17 + 0.111i)3-s + (−1.36 − 3.95i)5-s + (−0.966 − 2.46i)7-s + (−1.58 + 0.306i)9-s + (0.444 − 0.178i)11-s + (1.11 + 0.713i)13-s + (2.04 + 4.47i)15-s + (0.0590 + 0.243i)17-s + (−1.97 + 8.12i)19-s + (1.40 + 2.77i)21-s + (−0.234 + 4.79i)23-s + (−9.81 + 7.71i)25-s + (5.20 − 1.52i)27-s + (−8.65 − 2.54i)29-s + (5.37 − 7.55i)31-s + ⋯ |
| L(s) = 1 | + (−0.675 + 0.0645i)3-s + (−0.611 − 1.76i)5-s + (−0.365 − 0.930i)7-s + (−0.529 + 0.102i)9-s + (0.134 − 0.0536i)11-s + (0.307 + 0.197i)13-s + (0.527 + 1.15i)15-s + (0.0143 + 0.0589i)17-s + (−0.452 + 1.86i)19-s + (0.306 + 0.605i)21-s + (−0.0489 + 0.998i)23-s + (−1.96 + 1.54i)25-s + (1.00 − 0.294i)27-s + (−1.60 − 0.471i)29-s + (0.966 − 1.35i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0568846 + 0.169347i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0568846 + 0.169347i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.966 + 2.46i)T \) |
| 23 | \( 1 + (0.234 - 4.79i)T \) |
| good | 3 | \( 1 + (1.17 - 0.111i)T + (2.94 - 0.567i)T^{2} \) |
| 5 | \( 1 + (1.36 + 3.95i)T + (-3.93 + 3.09i)T^{2} \) |
| 11 | \( 1 + (-0.444 + 0.178i)T + (7.96 - 7.59i)T^{2} \) |
| 13 | \( 1 + (-1.11 - 0.713i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.0590 - 0.243i)T + (-15.1 + 7.78i)T^{2} \) |
| 19 | \( 1 + (1.97 - 8.12i)T + (-16.8 - 8.70i)T^{2} \) |
| 29 | \( 1 + (8.65 + 2.54i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-5.37 + 7.55i)T + (-10.1 - 29.2i)T^{2} \) |
| 37 | \( 1 + (-3.38 + 0.653i)T + (34.3 - 13.7i)T^{2} \) |
| 41 | \( 1 + (-2.55 + 2.95i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (1.31 - 2.86i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-1.20 + 2.08i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.28 + 3.24i)T + (30.7 + 43.1i)T^{2} \) |
| 59 | \( 1 + (-0.366 + 7.69i)T + (-58.7 - 5.60i)T^{2} \) |
| 61 | \( 1 + (12.0 + 1.14i)T + (59.8 + 11.5i)T^{2} \) |
| 67 | \( 1 + (9.16 - 7.20i)T + (15.7 - 65.1i)T^{2} \) |
| 71 | \( 1 + (-1.26 - 8.81i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.36 - 1.30i)T + (3.47 + 72.9i)T^{2} \) |
| 79 | \( 1 + (-0.832 + 0.428i)T + (45.8 - 64.3i)T^{2} \) |
| 83 | \( 1 + (1.40 + 1.62i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (2.51 + 3.53i)T + (-29.1 + 84.1i)T^{2} \) |
| 97 | \( 1 + (11.2 - 12.9i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−9.959260217676909706637350587933, −9.231398023503965317612777220320, −8.143772780415425604743708443789, −7.69344260748630428378627119737, −6.14011864900866935642115797737, −5.51325738788856833178360193845, −4.34943140727104489955076157190, −3.75520554695331912838325973331, −1.43382408908734368526486962727, −0.10789873456697812675966125142,
2.61270599724422607542659210605, 3.20393371923938140093852392238, 4.70921212830919264218889702613, 6.02058074757354153838606002743, 6.51198885482979329354085832510, 7.31385330608191009114610267069, 8.500278428074118208047699421841, 9.363815731954206680567174009773, 10.72904940213339691510311257685, 10.91257642297138113360550607812
