| L(s) = 1 | + (−7.12 − 2.94i)2-s + (−0.829 + 0.165i)3-s + (30.6 + 30.6i)4-s + (−14.8 + 22.2i)5-s + (6.39 + 1.27i)6-s + (16.8 + 25.2i)7-s + (−80.8 − 195. i)8-s + (−74.1 + 30.7i)9-s + (171. − 114. i)10-s + (−28.3 + 142. i)11-s + (−30.5 − 20.3i)12-s + (−15.1 + 15.1i)13-s + (−45.7 − 229. i)14-s + (8.66 − 20.9i)15-s + 933. i·16-s + (191. − 216. i)17-s + ⋯ |
| L(s) = 1 | + (−1.78 − 0.737i)2-s + (−0.0921 + 0.0183i)3-s + (1.91 + 1.91i)4-s + (−0.594 + 0.890i)5-s + (0.177 + 0.0353i)6-s + (0.344 + 0.516i)7-s + (−1.26 − 3.04i)8-s + (−0.915 + 0.379i)9-s + (1.71 − 1.14i)10-s + (−0.234 + 1.17i)11-s + (−0.211 − 0.141i)12-s + (−0.0897 + 0.0897i)13-s + (−0.233 − 1.17i)14-s + (0.0385 − 0.0929i)15-s + 3.64i·16-s + (0.661 − 0.749i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.265961 + 0.234855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.265961 + 0.234855i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-191. + 216. i)T \) |
| good | 2 | \( 1 + (7.12 + 2.94i)T + (11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (0.829 - 0.165i)T + (74.8 - 30.9i)T^{2} \) |
| 5 | \( 1 + (14.8 - 22.2i)T + (-239. - 577. i)T^{2} \) |
| 7 | \( 1 + (-16.8 - 25.2i)T + (-918. + 2.21e3i)T^{2} \) |
| 11 | \( 1 + (28.3 - 142. i)T + (-1.35e4 - 5.60e3i)T^{2} \) |
| 13 | \( 1 + (15.1 - 15.1i)T - 2.85e4iT^{2} \) |
| 19 | \( 1 + (190. + 79.0i)T + (9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (335. + 66.7i)T + (2.58e5 + 1.07e5i)T^{2} \) |
| 29 | \( 1 + (-923. - 617. i)T + (2.70e5 + 6.53e5i)T^{2} \) |
| 31 | \( 1 + (7.20 + 36.2i)T + (-8.53e5 + 3.53e5i)T^{2} \) |
| 37 | \( 1 + (-184. + 36.6i)T + (1.73e6 - 7.17e5i)T^{2} \) |
| 41 | \( 1 + (-926. - 1.38e3i)T + (-1.08e6 + 2.61e6i)T^{2} \) |
| 43 | \( 1 + (1.07e3 - 447. i)T + (2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (-668. + 668. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.14e3 - 473. i)T + (5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-1.50e3 - 3.63e3i)T + (-8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (-3.62e3 + 2.42e3i)T + (5.29e6 - 1.27e7i)T^{2} \) |
| 67 | \( 1 - 2.30e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (4.92e3 - 978. i)T + (2.34e7 - 9.72e6i)T^{2} \) |
| 73 | \( 1 + (2.47e3 - 3.70e3i)T + (-1.08e7 - 2.62e7i)T^{2} \) |
| 79 | \( 1 + (-187. + 940. i)T + (-3.59e7 - 1.49e7i)T^{2} \) |
| 83 | \( 1 + (4.59e3 - 1.10e4i)T + (-3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (-305. - 305. i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (-9.82e3 - 6.56e3i)T + (3.38e7 + 8.17e7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.42112363542893849852518842422, −17.63744032836750553041924230939, −16.27930258984054430258842862408, −14.90382751032736604210395739314, −12.15045292915663003595424657181, −11.24672080046103990170189362028, −10.04147446447383999206885244719, −8.423322242421306942592925245856, −7.16937845799535847755191534194, −2.62087598818458817265345927037,
0.58817291624607655439736828838, 5.95279697387076790953311550260, 7.989406979852136903064390717112, 8.693936327998515649568238632805, 10.47024488480401062159810886452, 11.77188691763361885015700658851, 14.38505764712481821229397677600, 15.85440572720160247804364969740, 16.77117044031225316156515863168, 17.54537443031310059156007739779
