| L(s) = 1 | + (2.24 + 5.41i)2-s + (12.0 − 8.04i)3-s + (−12.9 + 12.9i)4-s + (−33.5 − 6.67i)5-s + (70.5 + 47.1i)6-s + (−47.5 + 9.46i)7-s + (−12.8 − 5.32i)8-s + (49.2 − 118. i)9-s + (−39.1 − 196. i)10-s + (18.1 − 27.1i)11-s + (−51.8 + 260. i)12-s + (162. + 162. i)13-s + (−158. − 236. i)14-s + (−457. + 189. i)15-s + 212. i·16-s + (136. − 254. i)17-s + ⋯ |
| L(s) = 1 | + (0.560 + 1.35i)2-s + (1.33 − 0.893i)3-s + (−0.811 + 0.811i)4-s + (−1.34 − 0.266i)5-s + (1.96 + 1.30i)6-s + (−0.971 + 0.193i)7-s + (−0.200 − 0.0831i)8-s + (0.607 − 1.46i)9-s + (−0.391 − 1.96i)10-s + (0.149 − 0.224i)11-s + (−0.360 + 1.81i)12-s + (0.961 + 0.961i)13-s + (−0.806 − 1.20i)14-s + (−2.03 + 0.842i)15-s + 0.829i·16-s + (0.471 − 0.882i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.59252 + 0.797116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59252 + 0.797116i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-136. + 254. i)T \) |
| good | 2 | \( 1 + (-2.24 - 5.41i)T + (-11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (-12.0 + 8.04i)T + (30.9 - 74.8i)T^{2} \) |
| 5 | \( 1 + (33.5 + 6.67i)T + (577. + 239. i)T^{2} \) |
| 7 | \( 1 + (47.5 - 9.46i)T + (2.21e3 - 918. i)T^{2} \) |
| 11 | \( 1 + (-18.1 + 27.1i)T + (-5.60e3 - 1.35e4i)T^{2} \) |
| 13 | \( 1 + (-162. - 162. i)T + 2.85e4iT^{2} \) |
| 19 | \( 1 + (-40.4 - 97.7i)T + (-9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (496. + 331. i)T + (1.07e5 + 2.58e5i)T^{2} \) |
| 29 | \( 1 + (-25.9 + 130. i)T + (-6.53e5 - 2.70e5i)T^{2} \) |
| 31 | \( 1 + (-113. - 169. i)T + (-3.53e5 + 8.53e5i)T^{2} \) |
| 37 | \( 1 + (296. - 197. i)T + (7.17e5 - 1.73e6i)T^{2} \) |
| 41 | \( 1 + (1.16e3 - 231. i)T + (2.61e6 - 1.08e6i)T^{2} \) |
| 43 | \( 1 + (-970. + 2.34e3i)T + (-2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (-1.19e3 - 1.19e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (1.05e3 + 2.54e3i)T + (-5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (1.42e3 + 591. i)T + (8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (-1.01e3 - 5.11e3i)T + (-1.27e7 + 5.29e6i)T^{2} \) |
| 67 | \( 1 - 1.95e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-3.09e3 + 2.06e3i)T + (9.72e6 - 2.34e7i)T^{2} \) |
| 73 | \( 1 + (-3.53e3 - 703. i)T + (2.62e7 + 1.08e7i)T^{2} \) |
| 79 | \( 1 + (-5.41e3 + 8.09e3i)T + (-1.49e7 - 3.59e7i)T^{2} \) |
| 83 | \( 1 + (5.09e3 - 2.11e3i)T + (3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (-7.25e3 + 7.25e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (2.42e3 - 1.21e4i)T + (-8.17e7 - 3.38e7i)T^{2} \) |
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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
−18.62182155072647365244480800220, −16.33250287095048470061432820909, −15.69508049490476561711538602861, −14.35723143472776578755008557554, −13.46736629407768929856912774548, −12.13048658649682655792832160362, −8.825996166664148934106742067211, −7.78514853924032931470776065558, −6.61101957613644967419947278431, −3.76022610772082635901349121910,
3.26345268154853224380187672616, 3.91943838237783152791223598028, 8.009028921276483134004877531003, 9.768321557145801920633985251133, 10.89650209327371942270552989420, 12.50651211056424317792684238530, 13.76543036567021003637594070215, 15.19923945284804422268400065942, 16.03613085616222404035690012830, 18.90983927092961214037806062684
