L(s) = 1 | + (−1.41 + 1.41i)4-s + (−1.14 + 2.77i)9-s + (−5.18 − 3.46i)11-s + (1.77 + 1.77i)13-s − 4.00i·16-s + (−1.05 − 3.98i)17-s + (5.18 − 7.76i)23-s + (−4.61 − 1.91i)25-s + (−6.27 + 4.19i)31-s + (−2.29 − 5.54i)36-s + (2.49 + 12.5i)41-s + (2.50 − 6.05i)43-s + (12.2 − 2.43i)44-s + (−9.65 − 9.65i)47-s + (−6.46 + 2.67i)49-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)4-s + (−0.382 + 0.923i)9-s + (−1.56 − 1.04i)11-s + (0.493 + 0.493i)13-s − 1.00i·16-s + (−0.255 − 0.966i)17-s + (1.08 − 1.61i)23-s + (−0.923 − 0.382i)25-s + (−1.12 + 0.753i)31-s + (−0.382 − 0.923i)36-s + (0.389 + 1.95i)41-s + (0.382 − 0.923i)43-s + (1.84 − 0.366i)44-s + (−1.40 − 1.40i)47-s + (−0.923 + 0.382i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.106293 - 0.225895i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.106293 - 0.225895i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (1.05 + 3.98i)T \) |
| 43 | \( 1 + (-2.50 + 6.05i)T \) |
good | 2 | \( 1 + (1.41 - 1.41i)T^{2} \) |
| 3 | \( 1 + (1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (5.18 + 3.46i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.77 - 1.77i)T + 13iT^{2} \) |
| 19 | \( 1 + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-5.18 + 7.76i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (6.27 - 4.19i)T + (11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-2.49 - 12.5i)T + (-37.8 + 15.6i)T^{2} \) |
| 47 | \( 1 + (9.65 + 9.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.55 + 13.4i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (2.19 + 0.907i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 - 0.318iT - 67T^{2} \) |
| 71 | \( 1 + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-2.23 - 1.49i)T + (30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (6.05 - 2.50i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + 89iT^{2} \) |
| 97 | \( 1 + (9.90 + 1.96i)T + (89.6 + 37.1i)T^{2} \) |
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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
−10.11640718711456479170856355678, −9.001179274367136507546393392968, −8.309752351971606787516974653436, −7.79673089505412598802858097951, −6.61808590076380342954414974818, −5.27053897384241798980965972861, −4.78112713590824735092975668925, −3.36534670356337475727394104588, −2.49484570194487245473473578513, −0.12738536170039738841728891413,
1.63879845114971410305473469140, 3.24972002939365257576016959410, 4.36043033490189218450415857789, 5.47944474766573794351233762519, 5.95304780087840629401449470459, 7.33290339183507229806892355604, 8.135199108470660968959472786409, 9.248242876924729168401417144806, 9.676420824830287195912374557545, 10.68351815724866780521749302897