| L(s) = 1 | + (−1.26 − 1.54i)2-s + (−0.779 + 3.92i)4-s + 6.46·5-s − 11.6i·7-s + (7.05 − 3.77i)8-s + (−8.20 − 9.99i)10-s + 8.68i·11-s − 21.6·13-s + (−18.0 + 14.7i)14-s + (−14.7 − 6.11i)16-s − 26.7·17-s − 4.35i·19-s + (−5.03 + 25.3i)20-s + (13.4 − 11.0i)22-s − 15.9i·23-s + ⋯ |
| L(s) = 1 | + (−0.634 − 0.772i)2-s + (−0.194 + 0.980i)4-s + 1.29·5-s − 1.66i·7-s + (0.881 − 0.471i)8-s + (−0.820 − 0.999i)10-s + 0.789i·11-s − 1.66·13-s + (−1.28 + 1.05i)14-s + (−0.924 − 0.382i)16-s − 1.57·17-s − 0.229i·19-s + (−0.251 + 1.26i)20-s + (0.610 − 0.500i)22-s − 0.693i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6851806968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6851806968\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 + 1.54i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - 6.46T + 25T^{2} \) |
| 7 | \( 1 + 11.6iT - 49T^{2} \) |
| 11 | \( 1 - 8.68iT - 121T^{2} \) |
| 13 | \( 1 + 21.6T + 169T^{2} \) |
| 17 | \( 1 + 26.7T + 289T^{2} \) |
| 23 | \( 1 + 15.9iT - 529T^{2} \) |
| 29 | \( 1 - 27.3T + 841T^{2} \) |
| 31 | \( 1 + 37.0iT - 961T^{2} \) |
| 37 | \( 1 + 38.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 18.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 30.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.86iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 4.73iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 75.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 105. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 20.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 33.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 72.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 164.T + 9.40e3T^{2} \) |
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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
−9.869246905503225680482612924120, −9.416791478324378944336911420547, −8.182731799633082633513620555905, −7.11719001445902060437581819357, −6.69022858024773383913053871263, −4.85215158347943847812317345888, −4.24932564863075020778639983375, −2.63288681806544275005345584804, −1.79429754775107871423978599001, −0.27512076782534417971952440321,
1.84473229512227667456454614359, 2.63421071769089225096030471952, 4.94144127100360433915025501901, 5.47644342016555114226790286138, 6.28795100208903234221115077771, 7.07236703057290404878382132845, 8.498889957200482063167303014024, 8.892717682321126853086215114171, 9.681024990375787201490830134194, 10.34274617519310124787877542546
