| L(s) = 1 | + (−3.17 + 1.31i)2-s + (−0.505 + 2.54i)3-s + (−2.98 + 2.98i)4-s + (−30.6 + 20.4i)5-s + (−1.73 − 8.73i)6-s + (18.6 + 12.4i)7-s + (26.5 − 64.1i)8-s + (68.6 + 28.4i)9-s + (70.2 − 105. i)10-s + (−80.7 + 16.0i)11-s + (−6.07 − 9.09i)12-s + (65.4 + 65.4i)13-s + (−75.5 − 15.0i)14-s + (−36.5 − 88.2i)15-s + 170. i·16-s + (−164. + 237. i)17-s + ⋯ |
| L(s) = 1 | + (−0.792 + 0.328i)2-s + (−0.0562 + 0.282i)3-s + (−0.186 + 0.186i)4-s + (−1.22 + 0.818i)5-s + (−0.0482 − 0.242i)6-s + (0.381 + 0.254i)7-s + (0.414 − 1.00i)8-s + (0.847 + 0.350i)9-s + (0.702 − 1.05i)10-s + (−0.667 + 0.132i)11-s + (−0.0422 − 0.0631i)12-s + (0.387 + 0.387i)13-s + (−0.385 − 0.0767i)14-s + (−0.162 − 0.392i)15-s + 0.666i·16-s + (−0.569 + 0.822i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.198833 + 0.508715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.198833 + 0.508715i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (164. - 237. i)T \) |
| good | 2 | \( 1 + (3.17 - 1.31i)T + (11.3 - 11.3i)T^{2} \) |
| 3 | \( 1 + (0.505 - 2.54i)T + (-74.8 - 30.9i)T^{2} \) |
| 5 | \( 1 + (30.6 - 20.4i)T + (239. - 577. i)T^{2} \) |
| 7 | \( 1 + (-18.6 - 12.4i)T + (918. + 2.21e3i)T^{2} \) |
| 11 | \( 1 + (80.7 - 16.0i)T + (1.35e4 - 5.60e3i)T^{2} \) |
| 13 | \( 1 + (-65.4 - 65.4i)T + 2.85e4iT^{2} \) |
| 19 | \( 1 + (-97.7 + 40.4i)T + (9.21e4 - 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-106. - 537. i)T + (-2.58e5 + 1.07e5i)T^{2} \) |
| 29 | \( 1 + (786. + 1.17e3i)T + (-2.70e5 + 6.53e5i)T^{2} \) |
| 31 | \( 1 + (-1.78e3 - 354. i)T + (8.53e5 + 3.53e5i)T^{2} \) |
| 37 | \( 1 + (-2.77 + 13.9i)T + (-1.73e6 - 7.17e5i)T^{2} \) |
| 41 | \( 1 + (-4.12 - 2.75i)T + (1.08e6 + 2.61e6i)T^{2} \) |
| 43 | \( 1 + (1.17e3 + 484. i)T + (2.41e6 + 2.41e6i)T^{2} \) |
| 47 | \( 1 + (-2.31e3 - 2.31e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-3.75e3 + 1.55e3i)T + (5.57e6 - 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-1.87e3 + 4.53e3i)T + (-8.56e6 - 8.56e6i)T^{2} \) |
| 61 | \( 1 + (767. - 1.14e3i)T + (-5.29e6 - 1.27e7i)T^{2} \) |
| 67 | \( 1 - 6.12e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (1.10e3 - 5.55e3i)T + (-2.34e7 - 9.72e6i)T^{2} \) |
| 73 | \( 1 + (6.11e3 - 4.08e3i)T + (1.08e7 - 2.62e7i)T^{2} \) |
| 79 | \( 1 + (1.33e3 - 265. i)T + (3.59e7 - 1.49e7i)T^{2} \) |
| 83 | \( 1 + (-1.80e3 - 4.36e3i)T + (-3.35e7 + 3.35e7i)T^{2} \) |
| 89 | \( 1 + (-5.37e3 + 5.37e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (483. + 723. i)T + (-3.38e7 + 8.17e7i)T^{2} \) |
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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
−18.69400614412518729065415323455, −17.47284328565801582409117223525, −15.88059029978838111992837739748, −15.31033082360402065386936875285, −13.25758888302205104233352424885, −11.48447573692504095870022858421, −10.10836464567477248979118747625, −8.273027580290593777159945921689, −7.20729626139803974778764658325, −4.09342913201418063237680625529,
0.73269144160509988761913603252, 4.67680407619971430610147404596, 7.64038889003522274962881568369, 8.823410736020819845449578640162, 10.54919171163016159933114431548, 11.94404622031031074404416611500, 13.38953551881684708729568299470, 15.27546205348362155260259253138, 16.47726658767781843077396226427, 18.02589129501753476235922977162
