| L(s) = 1 | + (−5.69 + 2.35i)2-s + (1.49 − 7.52i)3-s + (15.5 − 15.5i)4-s + (30.3 − 20.3i)5-s + (9.21 + 46.3i)6-s + (−64.4 − 43.0i)7-s + (−14.0 + 33.9i)8-s + (20.5 + 8.49i)9-s + (−125. + 187. i)10-s + (14.7 − 2.93i)11-s + (−93.5 − 140. i)12-s + (1.97 + 1.97i)13-s + (468. + 93.2i)14-s + (−107. − 258. i)15-s + 124. i·16-s + (215. − 192. i)17-s + ⋯ |
| L(s) = 1 | + (−1.42 + 0.589i)2-s + (0.166 − 0.835i)3-s + (0.970 − 0.970i)4-s + (1.21 − 0.812i)5-s + (0.256 + 1.28i)6-s + (−1.31 − 0.879i)7-s + (−0.219 + 0.530i)8-s + (0.253 + 0.104i)9-s + (−1.25 + 1.87i)10-s + (0.121 − 0.0242i)11-s + (−0.649 − 0.972i)12-s + (0.0116 + 0.0116i)13-s + (2.39 + 0.475i)14-s + (−0.476 − 1.15i)15-s + 0.488i·16-s + (0.744 − 0.667i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.614452 - 0.327443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.614452 - 0.327443i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-215. + 192. i)T \) |
| good | 2 | \( 1 + (5.69 - 2.35i)T + (11.3 - 11.3i)T^{2} \) |
| 3 | \( 1 + (-1.49 + 7.52i)T + (-74.8 - 30.9i)T^{2} \) |
| 5 | \( 1 + (-30.3 + 20.3i)T + (239. - 577. i)T^{2} \) |
| 7 | \( 1 + (64.4 + 43.0i)T + (918. + 2.21e3i)T^{2} \) |
| 11 | \( 1 + (-14.7 + 2.93i)T + (1.35e4 - 5.60e3i)T^{2} \) |
| 13 | \( 1 + (-1.97 - 1.97i)T + 2.85e4iT^{2} \) |
| 19 | \( 1 + (248. - 102. i)T + (9.21e4 - 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-183. - 923. i)T + (-2.58e5 + 1.07e5i)T^{2} \) |
| 29 | \( 1 + (-55.8 - 83.6i)T + (-2.70e5 + 6.53e5i)T^{2} \) |
| 31 | \( 1 + (-832. - 165. i)T + (8.53e5 + 3.53e5i)T^{2} \) |
| 37 | \( 1 + (-65.3 + 328. i)T + (-1.73e6 - 7.17e5i)T^{2} \) |
| 41 | \( 1 + (-2.20e3 - 1.47e3i)T + (1.08e6 + 2.61e6i)T^{2} \) |
| 43 | \( 1 + (-221. - 91.9i)T + (2.41e6 + 2.41e6i)T^{2} \) |
| 47 | \( 1 + (902. + 902. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (1.54e3 - 639. i)T + (5.57e6 - 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-30.9 + 74.6i)T + (-8.56e6 - 8.56e6i)T^{2} \) |
| 61 | \( 1 + (1.44e3 - 2.16e3i)T + (-5.29e6 - 1.27e7i)T^{2} \) |
| 67 | \( 1 + 4.40e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-544. + 2.73e3i)T + (-2.34e7 - 9.72e6i)T^{2} \) |
| 73 | \( 1 + (3.09e3 - 2.06e3i)T + (1.08e7 - 2.62e7i)T^{2} \) |
| 79 | \( 1 + (3.70e3 - 736. i)T + (3.59e7 - 1.49e7i)T^{2} \) |
| 83 | \( 1 + (-2.06e3 - 4.98e3i)T + (-3.35e7 + 3.35e7i)T^{2} \) |
| 89 | \( 1 + (-7.56e3 + 7.56e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (-5.33e3 - 7.98e3i)T + (-3.38e7 + 8.17e7i)T^{2} \) |
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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
−17.78734993375560541411177641403, −16.90080966247904131033176703277, −16.04436780294855127100088708878, −13.65855512720457745144419025881, −12.84630312197188110742282151771, −10.10242420153054099603168538804, −9.326110194225861268328825528655, −7.57066719963252819239842093584, −6.31308830120968832761487689234, −1.15799625279579921259712196806,
2.68441472778940780794188776055, 6.39489853255290170686219281128, 8.948646765982291228018661315309, 9.885553836097459960084091727960, 10.54590758658948127045055435233, 12.63059657687459581186909769987, 14.64368494633080213745766950673, 16.09240681119032640502340009672, 17.29901291659849314835975126695, 18.56904713935040240057607592061
