| L(s) = 1 | + (−1.37 − 2.37i)5-s + (−2.60 + 0.463i)7-s + (0.362 + 0.209i)11-s + (1.32 + 0.765i)13-s + (1.95 + 3.38i)17-s + (−5.11 − 2.95i)19-s + (−7.72 + 4.46i)23-s + (−1.26 + 2.18i)25-s + (−6.00 + 3.46i)29-s + 3.52i·31-s + (4.67 + 5.55i)35-s + (−4.54 + 7.87i)37-s + (−1.06 + 1.84i)41-s + (−5.77 − 10.0i)43-s + 1.77·47-s + ⋯ |
| L(s) = 1 | + (−0.613 − 1.06i)5-s + (−0.984 + 0.175i)7-s + (0.109 + 0.0630i)11-s + (0.367 + 0.212i)13-s + (0.473 + 0.820i)17-s + (−1.17 − 0.678i)19-s + (−1.61 + 0.930i)23-s + (−0.252 + 0.437i)25-s + (−1.11 + 0.643i)29-s + 0.633i·31-s + (0.790 + 0.938i)35-s + (−0.747 + 1.29i)37-s + (−0.165 + 0.287i)41-s + (−0.881 − 1.52i)43-s + 0.258·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0246486 + 0.0884930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0246486 + 0.0884930i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.60 - 0.463i)T \) |
| good | 5 | \( 1 + (1.37 + 2.37i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.362 - 0.209i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.32 - 0.765i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.95 - 3.38i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.72 - 4.46i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.00 - 3.46i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.52iT - 31T^{2} \) |
| 37 | \( 1 + (4.54 - 7.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.06 - 1.84i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.77 + 10.0i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 1.77T + 47T^{2} \) |
| 53 | \( 1 + (-3.39 + 1.96i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 4.05T + 59T^{2} \) |
| 61 | \( 1 + 1.86iT - 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 8.51iT - 71T^{2} \) |
| 73 | \( 1 + (-1.65 + 0.952i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 0.867T + 79T^{2} \) |
| 83 | \( 1 + (3.45 + 5.99i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.88 + 8.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.200 + 0.115i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.57166449343121964261460989885, −9.820164759449154867008721122932, −8.775777306389752377971516103748, −8.423701775399005813245773902318, −7.23612149701467294460801976932, −6.27338239164400788402862574005, −5.35608251560232463742963412561, −4.18483052608709377008715273936, −3.45076560940656968432830323768, −1.72987262462237872673986787430,
0.04468108036543870806264548219, 2.35474053596196699295863096311, 3.48698921681481288169365574349, 4.13770435071900112106149794129, 5.82487891388025570727766283384, 6.46694250406839844780784766304, 7.37290252770011779935781590250, 8.107492618573316636374085855616, 9.250976127928256674945127215799, 10.16547443417918901810681943849
