| L(s) = 1 | + (−1.66 − 4.01i)2-s + (−14.6 + 9.80i)3-s + (−2.05 + 2.05i)4-s + (−17.1 − 3.41i)5-s + (63.7 + 42.6i)6-s + (−18.6 + 3.70i)7-s + (−52.5 − 21.7i)8-s + (88.1 − 212. i)9-s + (14.8 + 74.6i)10-s + (−48.3 + 72.2i)11-s + (10.0 − 50.3i)12-s + (74.9 + 74.9i)13-s + (45.9 + 68.7i)14-s + (285. − 118. i)15-s + 294. i·16-s + (−274. − 90.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.416 − 1.00i)2-s + (−1.62 + 1.08i)3-s + (−0.128 + 0.128i)4-s + (−0.686 − 0.136i)5-s + (1.77 + 1.18i)6-s + (−0.380 + 0.0756i)7-s + (−0.821 − 0.340i)8-s + (1.08 − 2.62i)9-s + (0.148 + 0.746i)10-s + (−0.399 + 0.597i)11-s + (0.0695 − 0.349i)12-s + (0.443 + 0.443i)13-s + (0.234 + 0.350i)14-s + (1.26 − 0.524i)15-s + 1.14i·16-s + (−0.949 − 0.312i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0101220 + 0.0422848i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0101220 + 0.0422848i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (274. + 90.2i)T \) |
| good | 2 | \( 1 + (1.66 + 4.01i)T + (-11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (14.6 - 9.80i)T + (30.9 - 74.8i)T^{2} \) |
| 5 | \( 1 + (17.1 + 3.41i)T + (577. + 239. i)T^{2} \) |
| 7 | \( 1 + (18.6 - 3.70i)T + (2.21e3 - 918. i)T^{2} \) |
| 11 | \( 1 + (48.3 - 72.2i)T + (-5.60e3 - 1.35e4i)T^{2} \) |
| 13 | \( 1 + (-74.9 - 74.9i)T + 2.85e4iT^{2} \) |
| 19 | \( 1 + (-37.3 - 90.2i)T + (-9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (457. + 305. i)T + (1.07e5 + 2.58e5i)T^{2} \) |
| 29 | \( 1 + (-172. + 865. i)T + (-6.53e5 - 2.70e5i)T^{2} \) |
| 31 | \( 1 + (-7.47 - 11.1i)T + (-3.53e5 + 8.53e5i)T^{2} \) |
| 37 | \( 1 + (186. - 124. i)T + (7.17e5 - 1.73e6i)T^{2} \) |
| 41 | \( 1 + (-376. + 74.8i)T + (2.61e6 - 1.08e6i)T^{2} \) |
| 43 | \( 1 + (-228. + 552. i)T + (-2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (-1.27e3 - 1.27e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-803. - 1.93e3i)T + (-5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (3.13e3 + 1.29e3i)T + (8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (368. + 1.85e3i)T + (-1.27e7 + 5.29e6i)T^{2} \) |
| 67 | \( 1 + 2.68e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (3.85e3 - 2.57e3i)T + (9.72e6 - 2.34e7i)T^{2} \) |
| 73 | \( 1 + (-2.14e3 - 427. i)T + (2.62e7 + 1.08e7i)T^{2} \) |
| 79 | \( 1 + (-2.65e3 + 3.97e3i)T + (-1.49e7 - 3.59e7i)T^{2} \) |
| 83 | \( 1 + (4.52e3 - 1.87e3i)T + (3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (-1.77e3 + 1.77e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (-3.07e3 + 1.54e4i)T + (-8.17e7 - 3.38e7i)T^{2} \) |
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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
−17.54199279192271564770517882341, −16.02048660567492593997810393944, −15.45256342989825204180171887857, −12.38451129107033946182419853210, −11.56280344237601765776599452389, −10.54317046505225755750064302141, −9.461870823878423114313932591371, −6.25878749744949906310613284182, −4.21445501236246627964536681282, −0.05470951818060648376561345048,
5.73046654250902156502322949471, 6.89080771366204891127762601112, 8.045308235695239383055341129873, 10.92068179235526547813644928509, 11.98919176366260081529261061090, 13.31323889023236822891597048374, 15.69224002402803375831090298560, 16.37148382639898533525538639126, 17.58753867102276112211076108660, 18.29632891251243994580832083740
