| L(s) = 1 | + (−1.12 − 0.349i)2-s + (−0.626 − 0.281i)3-s + (−0.510 − 0.352i)4-s + (1.91 + 0.115i)5-s + (0.603 + 0.535i)6-s + (4.39 − 2.30i)7-s + (1.89 + 2.42i)8-s + (−1.67 − 1.89i)9-s + (−2.10 − 0.798i)10-s + (2.13 + 2.54i)11-s + (0.220 + 0.364i)12-s + (0.105 − 0.0730i)13-s + (−5.73 + 1.05i)14-s + (−1.16 − 0.611i)15-s + (−0.842 − 2.22i)16-s + (0.698 + 5.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.793 − 0.247i)2-s + (−0.361 − 0.162i)3-s + (−0.255 − 0.176i)4-s + (0.855 + 0.0517i)5-s + (0.246 + 0.218i)6-s + (1.65 − 0.871i)7-s + (0.671 + 0.856i)8-s + (−0.558 − 0.630i)9-s + (−0.665 − 0.252i)10-s + (0.642 + 0.765i)11-s + (0.0636 + 0.105i)12-s + (0.0293 − 0.0202i)13-s + (−1.53 + 0.280i)14-s + (−0.300 − 0.157i)15-s + (−0.210 − 0.555i)16-s + (0.169 + 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.983525 - 0.468129i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.983525 - 0.468129i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-2.13 - 2.54i)T \) |
| 53 | \( 1 + (-0.925 + 7.22i)T \) |
| good | 2 | \( 1 + (1.12 + 0.349i)T + (1.64 + 1.13i)T^{2} \) |
| 3 | \( 1 + (0.626 + 0.281i)T + (1.98 + 2.24i)T^{2} \) |
| 5 | \( 1 + (-1.91 - 0.115i)T + (4.96 + 0.602i)T^{2} \) |
| 7 | \( 1 + (-4.39 + 2.30i)T + (3.97 - 5.76i)T^{2} \) |
| 13 | \( 1 + (-0.105 + 0.0730i)T + (4.60 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.698 - 5.75i)T + (-16.5 + 4.06i)T^{2} \) |
| 19 | \( 1 + (-0.0592 + 0.323i)T + (-17.7 - 6.73i)T^{2} \) |
| 23 | \( 1 + (-2.27 - 2.27i)T + 23iT^{2} \) |
| 29 | \( 1 + (-7.97 + 1.96i)T + (25.6 - 13.4i)T^{2} \) |
| 31 | \( 1 + (-1.45 + 2.40i)T + (-14.4 - 27.4i)T^{2} \) |
| 37 | \( 1 + (-0.312 + 0.118i)T + (27.6 - 24.5i)T^{2} \) |
| 41 | \( 1 + (3.45 + 5.71i)T + (-19.0 + 36.3i)T^{2} \) |
| 43 | \( 1 + (0.623 - 1.64i)T + (-32.1 - 28.5i)T^{2} \) |
| 47 | \( 1 + (1.24 + 1.10i)T + (5.66 + 46.6i)T^{2} \) |
| 59 | \( 1 + (-4.35 + 4.91i)T + (-7.11 - 58.5i)T^{2} \) |
| 61 | \( 1 + (-3.99 - 5.10i)T + (-14.5 + 59.2i)T^{2} \) |
| 67 | \( 1 + (3.87 - 0.709i)T + (62.6 - 23.7i)T^{2} \) |
| 71 | \( 1 + (12.3 - 5.55i)T + (47.0 - 53.1i)T^{2} \) |
| 73 | \( 1 + (0.240 - 0.307i)T + (-17.4 - 70.8i)T^{2} \) |
| 79 | \( 1 + (0.430 + 1.38i)T + (-65.0 + 44.8i)T^{2} \) |
| 83 | \( 1 + (2.05 + 2.05i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.492 + 4.05i)T + (-86.4 + 21.2i)T^{2} \) |
| 97 | \( 1 + (1.52 - 1.35i)T + (11.6 - 96.2i)T^{2} \) |
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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
−10.38183438211231979386055328228, −9.916685974611269422084512205957, −8.793108384350962219373269329449, −8.201613633469368109008625158724, −7.12761196935221818603213847318, −5.98933565305083958801077460940, −5.03649355312512718661611064334, −4.08543766394651258891970864354, −1.92960716984901161511314171193, −1.12684165535396952934868035605,
1.26600789088806165744621348945, 2.72108810615936625208464087308, 4.64835479850893612615855375115, 5.22213318553581950753513722123, 6.25890775831300431808713756670, 7.54188769223615499789032075562, 8.517712364411115347489795628809, 8.791984139070014852264527873474, 9.823499097459389965560789551422, 10.76366811380078497851001103257
