| L(s) = 1 | + (0.439 − 2.49i)2-s + (0.5 + 0.419i)3-s + (−4.14 − 1.50i)4-s + (1.26 − 0.460i)5-s + (1.26 − 1.06i)6-s + (−0.766 − 1.32i)7-s + (−3.05 + 5.28i)8-s + (−0.446 − 2.53i)9-s + (−0.592 − 3.35i)10-s + (0.592 − 1.02i)11-s + (−1.43 − 2.49i)12-s + (−2.08 + 1.74i)13-s + (−3.64 + 1.32i)14-s + (0.826 + 0.300i)15-s + (5.08 + 4.26i)16-s + (0.673 − 3.82i)17-s + ⋯ |
| L(s) = 1 | + (0.310 − 1.76i)2-s + (0.288 + 0.242i)3-s + (−2.07 − 0.754i)4-s + (0.566 − 0.206i)5-s + (0.516 − 0.433i)6-s + (−0.289 − 0.501i)7-s + (−1.07 + 1.86i)8-s + (−0.148 − 0.844i)9-s + (−0.187 − 1.06i)10-s + (0.178 − 0.309i)11-s + (−0.415 − 0.719i)12-s + (−0.577 + 0.484i)13-s + (−0.974 + 0.354i)14-s + (0.213 + 0.0776i)15-s + (1.27 + 1.06i)16-s + (0.163 − 0.926i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0328023 - 1.49873i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0328023 - 1.49873i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-0.439 + 2.49i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (-0.5 - 0.419i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-1.26 + 0.460i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.766 + 1.32i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.592 + 1.02i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.08 - 1.74i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.673 + 3.82i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-4.75 - 1.73i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.807 - 4.58i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.91 + 3.32i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.10T + 37T^{2} \) |
| 41 | \( 1 + (-7.64 - 6.41i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.17 + 2.97i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.0996 - 0.565i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.76 - 1.00i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.683 + 3.87i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.24 - 1.54i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.674 - 3.82i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (6.51 - 2.37i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.69 - 3.93i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (7.51 + 6.30i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.15 + 10.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.85 + 1.55i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.27 - 7.25i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.13942657093225697979253738978, −10.03150558034960373081767895411, −9.430815958002629642313832710457, −8.937663844815267492450287607555, −7.19128790031534900923901703145, −5.71805625345296355534815141945, −4.53843384398940446817910990877, −3.54790212358881748636444212302, −2.55956721922193692350991267859, −0.964872059131422950023647900775,
2.53823856851150391309154443011, 4.29405709895564024866149913812, 5.44499116907008480061602180093, 6.08668506565689156588123779626, 7.15626523831891339619030481470, 7.910401426281966836066245125371, 8.770393215805593517256442904550, 9.670826001954374711438876044074, 10.78395853192284932424881207322, 12.45736347390780316422687505244
