| L(s) = 1 | + (−1.18 + 2.50i)2-s + (2.25 + 2.08i)3-s + (−3.58 − 4.38i)4-s + (1.92 + 1.13i)5-s + (−7.88 + 3.17i)6-s + (−0.850 − 0.246i)7-s + (9.85 − 2.42i)8-s + (0.516 + 6.39i)9-s + (−5.11 + 3.48i)10-s + (2.54 + 2.99i)11-s + (1.04 − 17.3i)12-s + (−3.28 − 1.49i)13-s + (1.62 − 1.83i)14-s + (2.00 + 6.56i)15-s + (−3.35 + 16.4i)16-s + (2.27 + 4.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.839 + 1.76i)2-s + (1.30 + 1.20i)3-s + (−1.79 − 2.19i)4-s + (0.862 + 0.505i)5-s + (−3.21 + 1.29i)6-s + (−0.321 − 0.0931i)7-s + (3.48 − 0.858i)8-s + (0.172 + 2.13i)9-s + (−1.61 + 1.10i)10-s + (0.768 + 0.903i)11-s + (0.303 − 5.01i)12-s + (−0.910 − 0.414i)13-s + (0.434 − 0.490i)14-s + (0.516 + 1.69i)15-s + (−0.838 + 4.10i)16-s + (0.551 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.519411 - 1.49630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.519411 - 1.49630i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.92 - 1.13i)T \) |
| 13 | \( 1 + (3.28 + 1.49i)T \) |
| good | 2 | \( 1 + (1.18 - 2.50i)T + (-1.26 - 1.54i)T^{2} \) |
| 3 | \( 1 + (-2.25 - 2.08i)T + (0.241 + 2.99i)T^{2} \) |
| 7 | \( 1 + (0.850 + 0.246i)T + (5.91 + 3.74i)T^{2} \) |
| 11 | \( 1 + (-2.54 - 2.99i)T + (-1.76 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-2.27 - 4.13i)T + (-9.08 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-1.16 + 0.312i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.841 + 3.13i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-5.98 - 2.84i)T + (18.3 + 22.4i)T^{2} \) |
| 31 | \( 1 + (2.96 + 2.32i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (3.97 + 5.29i)T + (-10.2 + 35.5i)T^{2} \) |
| 41 | \( 1 + (2.95 + 2.72i)T + (3.29 + 40.8i)T^{2} \) |
| 43 | \( 1 + (1.30 + 9.16i)T + (-41.3 + 11.9i)T^{2} \) |
| 47 | \( 1 + (4.34 - 1.64i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (-3.45 + 5.71i)T + (-24.6 - 46.9i)T^{2} \) |
| 59 | \( 1 + (-0.583 - 0.385i)T + (23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (-0.990 + 1.03i)T + (-2.45 - 60.9i)T^{2} \) |
| 67 | \( 1 + (2.05 - 2.51i)T + (-13.4 - 65.6i)T^{2} \) |
| 71 | \( 1 + (0.406 - 1.80i)T + (-64.1 - 30.4i)T^{2} \) |
| 73 | \( 1 + (5.75 - 8.33i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-13.6 + 5.16i)T + (59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (-4.26 - 8.11i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (4.18 + 1.12i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-13.4 + 4.49i)T + (77.5 - 58.2i)T^{2} \) |
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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
−10.13158106030801925740148087282, −9.677439653658894943591532640753, −8.978615818442370128415850096223, −8.324114249735605467312778015262, −7.32510482452992832784239717367, −6.69872432091944527949289070439, −5.50624493260948637802404615194, −4.74577619407734310473787821484, −3.63505114884547881157135156981, −1.98834618901601006441358635623,
0.956721935849290890943805755299, 1.74905535684331866165217343522, 2.81294480337455570749872906772, 3.33484454842719626014663878718, 4.81284087439805998630090353873, 6.55325809545963098324135541569, 7.61758575486327882451111158931, 8.345691677021042872944587437117, 9.168081104954281068566496938075, 9.422028936224148025779499918698
