| L(s) = 1 | + (25.0 + 5.72i)2-s + (18.6 + 38.6i)3-s + (135. + 65.1i)4-s + (−277. + 1.21e3i)5-s + (245. + 1.07e3i)6-s + (−7.76e3 + 3.73e3i)7-s + (−7.27e3 − 5.80e3i)8-s + (1.11e4 − 1.39e4i)9-s + (−1.39e4 + 2.89e4i)10-s + (−4.59e4 + 3.66e4i)11-s + 6.44e3i·12-s + (1.81e4 + 2.28e4i)13-s + (−2.16e5 + 4.93e4i)14-s + (−5.22e4 + 1.19e4i)15-s + (−1.97e5 − 2.47e5i)16-s + 3.74e5i·17-s + ⋯ |
| L(s) = 1 | + (1.10 + 0.253i)2-s + (0.132 + 0.275i)3-s + (0.264 + 0.127i)4-s + (−0.198 + 0.871i)5-s + (0.0774 + 0.339i)6-s + (−1.22 + 0.588i)7-s + (−0.628 − 0.501i)8-s + (0.565 − 0.708i)9-s + (−0.440 + 0.915i)10-s + (−0.946 + 0.754i)11-s + 0.0897i·12-s + (0.176 + 0.221i)13-s + (−1.50 + 0.343i)14-s + (−0.266 + 0.0608i)15-s + (−0.752 − 0.943i)16-s + 1.08i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.234599 + 1.45879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.234599 + 1.45879i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-9.88e5 - 3.67e6i)T \) |
| good | 2 | \( 1 + (-25.0 - 5.72i)T + (461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-18.6 - 38.6i)T + (-1.22e4 + 1.53e4i)T^{2} \) |
| 5 | \( 1 + (277. - 1.21e3i)T + (-1.75e6 - 8.47e5i)T^{2} \) |
| 7 | \( 1 + (7.76e3 - 3.73e3i)T + (2.51e7 - 3.15e7i)T^{2} \) |
| 11 | \( 1 + (4.59e4 - 3.66e4i)T + (5.24e8 - 2.29e9i)T^{2} \) |
| 13 | \( 1 + (-1.81e4 - 2.28e4i)T + (-2.35e9 + 1.03e10i)T^{2} \) |
| 17 | \( 1 - 3.74e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + (8.22e4 - 1.70e5i)T + (-2.01e11 - 2.52e11i)T^{2} \) |
| 23 | \( 1 + (1.72e5 + 7.55e5i)T + (-1.62e12 + 7.81e11i)T^{2} \) |
| 31 | \( 1 + (9.75e6 + 2.22e6i)T + (2.38e13 + 1.14e13i)T^{2} \) |
| 37 | \( 1 + (-1.63e7 - 1.30e7i)T + (2.89e13 + 1.26e14i)T^{2} \) |
| 41 | \( 1 - 1.53e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (-3.77e7 + 8.62e6i)T + (4.52e14 - 2.18e14i)T^{2} \) |
| 47 | \( 1 + (-2.46e7 + 1.96e7i)T + (2.49e14 - 1.09e15i)T^{2} \) |
| 53 | \( 1 + (4.57e5 - 2.00e6i)T + (-2.97e15 - 1.43e15i)T^{2} \) |
| 59 | \( 1 + 7.44e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + (4.82e6 + 1.00e7i)T + (-7.29e15 + 9.14e15i)T^{2} \) |
| 67 | \( 1 + (1.19e8 - 1.49e8i)T + (-6.05e15 - 2.65e16i)T^{2} \) |
| 71 | \( 1 + (3.91e7 + 4.90e7i)T + (-1.02e16 + 4.46e16i)T^{2} \) |
| 73 | \( 1 + (3.04e8 - 6.93e7i)T + (5.30e16 - 2.55e16i)T^{2} \) |
| 79 | \( 1 + (-3.44e8 - 2.74e8i)T + (2.66e16 + 1.16e17i)T^{2} \) |
| 83 | \( 1 + (6.33e8 + 3.04e8i)T + (1.16e17 + 1.46e17i)T^{2} \) |
| 89 | \( 1 + (-3.78e7 - 8.64e6i)T + (3.15e17 + 1.52e17i)T^{2} \) |
| 97 | \( 1 + (6.29e8 - 1.30e9i)T + (-4.73e17 - 5.94e17i)T^{2} \) |
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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
−15.20924368853657464084670376141, −14.67929121439363102194563880357, −12.97702883800128179454052692065, −12.44917524876437833753939577508, −10.44488848303438286278978227028, −9.271211919721612402290179968828, −6.98204108088057782060896809780, −5.91148565109723176009231795030, −4.07523442758987728112330492485, −2.91196992521110401284119264214,
0.39320055327780845852389170507, 2.83288034563639135715547921733, 4.32447909762814163043303390842, 5.68199413099746776332924471420, 7.59046301886010943241867550179, 9.209509890111155500726993305998, 10.92212275626251343409310934009, 12.61197137717552160522206774465, 13.13858383766890856265578559598, 13.91723729329751906293960543844
