| L(s) = 1 | + (5.21 − 2.15i)2-s + (−0.281 + 1.41i)3-s + (11.2 − 11.2i)4-s + (−9.99 + 6.67i)5-s + (1.58 + 7.97i)6-s + (−35.0 − 23.4i)7-s + (−0.333 + 0.805i)8-s + (72.9 + 30.2i)9-s + (−37.6 + 56.3i)10-s + (−29.4 + 5.86i)11-s + (12.6 + 18.9i)12-s + (−75.4 − 75.4i)13-s + (−233. − 46.3i)14-s + (−6.62 − 16.0i)15-s + 258. i·16-s + (87.2 − 275. i)17-s + ⋯ |
| L(s) = 1 | + (1.30 − 0.539i)2-s + (−0.0312 + 0.157i)3-s + (0.700 − 0.700i)4-s + (−0.399 + 0.267i)5-s + (0.0440 + 0.221i)6-s + (−0.715 − 0.477i)7-s + (−0.00521 + 0.0125i)8-s + (0.900 + 0.372i)9-s + (−0.376 + 0.563i)10-s + (−0.243 + 0.0484i)11-s + (0.0881 + 0.131i)12-s + (−0.446 − 0.446i)13-s + (−1.19 − 0.236i)14-s + (−0.0294 − 0.0711i)15-s + 1.00i·16-s + (0.302 − 0.953i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.431i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.90173 - 0.431827i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90173 - 0.431827i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-87.2 + 275. i)T \) |
| good | 2 | \( 1 + (-5.21 + 2.15i)T + (11.3 - 11.3i)T^{2} \) |
| 3 | \( 1 + (0.281 - 1.41i)T + (-74.8 - 30.9i)T^{2} \) |
| 5 | \( 1 + (9.99 - 6.67i)T + (239. - 577. i)T^{2} \) |
| 7 | \( 1 + (35.0 + 23.4i)T + (918. + 2.21e3i)T^{2} \) |
| 11 | \( 1 + (29.4 - 5.86i)T + (1.35e4 - 5.60e3i)T^{2} \) |
| 13 | \( 1 + (75.4 + 75.4i)T + 2.85e4iT^{2} \) |
| 19 | \( 1 + (-542. + 224. i)T + (9.21e4 - 9.21e4i)T^{2} \) |
| 23 | \( 1 + (64.9 + 326. i)T + (-2.58e5 + 1.07e5i)T^{2} \) |
| 29 | \( 1 + (-219. - 328. i)T + (-2.70e5 + 6.53e5i)T^{2} \) |
| 31 | \( 1 + (1.33e3 + 265. i)T + (8.53e5 + 3.53e5i)T^{2} \) |
| 37 | \( 1 + (439. - 2.20e3i)T + (-1.73e6 - 7.17e5i)T^{2} \) |
| 41 | \( 1 + (-1.86e3 - 1.24e3i)T + (1.08e6 + 2.61e6i)T^{2} \) |
| 43 | \( 1 + (1.90e3 + 789. i)T + (2.41e6 + 2.41e6i)T^{2} \) |
| 47 | \( 1 + (46.4 + 46.4i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.55e3 + 643. i)T + (5.57e6 - 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-901. + 2.17e3i)T + (-8.56e6 - 8.56e6i)T^{2} \) |
| 61 | \( 1 + (-31.9 + 47.8i)T + (-5.29e6 - 1.27e7i)T^{2} \) |
| 67 | \( 1 - 3.79e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-1.59e3 + 8.03e3i)T + (-2.34e7 - 9.72e6i)T^{2} \) |
| 73 | \( 1 + (-341. + 228. i)T + (1.08e7 - 2.62e7i)T^{2} \) |
| 79 | \( 1 + (1.94e3 - 387. i)T + (3.59e7 - 1.49e7i)T^{2} \) |
| 83 | \( 1 + (2.70e3 + 6.54e3i)T + (-3.35e7 + 3.35e7i)T^{2} \) |
| 89 | \( 1 + (6.49e3 - 6.49e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (8.98e3 + 1.34e4i)T + (-3.38e7 + 8.17e7i)T^{2} \) |
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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
−18.28624311846760733687145397335, −16.33769325323904605293913128254, −15.19157922892073407007238549665, −13.74188985019421153428349223486, −12.79511122707142210131829742246, −11.43377416120146748707380679772, −9.956118649384833467742771131290, −7.25673045693024397981774638278, −5.00327772393835329367043625035, −3.27944951518432818652421811640,
3.84459269239763967201718063418, 5.73527463902607383879088492791, 7.32469660623350521284503796682, 9.659112404222538856259969811677, 12.11897691919250085710310248485, 12.82555012569891124328751882848, 14.24885573041051237683615596077, 15.57852165958355091469542535219, 16.26864710285434519040228699792, 18.29039024826701433571817285042
