| L(s) = 1 | + (−4.45 + 2.97i)3-s + (2.23 + 0.444i)5-s + (−9.31 + 1.85i)7-s + (7.55 − 18.2i)9-s + (8.39 − 12.5i)11-s + (−8.16 − 8.16i)13-s + (−11.2 + 4.67i)15-s + (−11.0 + 12.8i)17-s + (−8.53 − 20.5i)19-s + (36.0 − 36.0i)21-s + (−11.2 − 7.54i)23-s + (−18.3 − 7.58i)25-s + (11.2 + 56.4i)27-s + (−1.19 + 6.01i)29-s + (25.4 + 38.0i)31-s + ⋯ |
| L(s) = 1 | + (−1.48 + 0.992i)3-s + (0.446 + 0.0888i)5-s + (−1.33 + 0.264i)7-s + (0.839 − 2.02i)9-s + (0.763 − 1.14i)11-s + (−0.628 − 0.628i)13-s + (−0.751 + 0.311i)15-s + (−0.652 + 0.757i)17-s + (−0.448 − 1.08i)19-s + (1.71 − 1.71i)21-s + (−0.490 − 0.327i)23-s + (−0.732 − 0.303i)25-s + (0.415 + 2.09i)27-s + (−0.0412 + 0.207i)29-s + (0.820 + 1.22i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0875600 - 0.142724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0875600 - 0.142724i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (11.0 - 12.8i)T \) |
| good | 3 | \( 1 + (4.45 - 2.97i)T + (3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (-2.23 - 0.444i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (9.31 - 1.85i)T + (45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-8.39 + 12.5i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (8.16 + 8.16i)T + 169iT^{2} \) |
| 19 | \( 1 + (8.53 + 20.5i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (11.2 + 7.54i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (1.19 - 6.01i)T + (-776. - 321. i)T^{2} \) |
| 31 | \( 1 + (-25.4 - 38.0i)T + (-367. + 887. i)T^{2} \) |
| 37 | \( 1 + (-10.3 + 6.90i)T + (523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (47.7 - 9.49i)T + (1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (15.3 - 36.9i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (16.7 + 16.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (22.0 + 53.2i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (73.9 + 30.6i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-8.02 - 40.3i)T + (-3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 + 105. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (96.0 - 64.1i)T + (1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (56.4 + 11.2i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-36.6 + 54.8i)T + (-2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-84.6 + 35.0i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-19.1 + 19.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (8.61 - 43.2i)T + (-8.69e3 - 3.60e3i)T^{2} \) |
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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
−12.45299865796529732993323085293, −11.52312959385346641629408241617, −10.51871429409293255250951519892, −9.840166156297918309180863838327, −8.826722863492630315776523908909, −6.44005437872886187867118988283, −6.18262666836984020897242710757, −4.81499249333911891031643794452, −3.36613223403990322482071002666, −0.12802752842557937386401729523,
1.88020015532886917678563652507, 4.42563678788579783074488265175, 5.93329407797104197192863146084, 6.64357427985425713299998867272, 7.45320823268628082309491185788, 9.524972870192702027787498495923, 10.18138795808730751079516911166, 11.67357411053056193247632176454, 12.19171599223074710661940837226, 13.09371439487442571618680033709
