| L(s) = 1 | + (6.53 + 2.70i)2-s + (−10.8 + 2.15i)3-s + (24.0 + 24.0i)4-s + (16.7 − 25.0i)5-s + (−76.7 − 15.2i)6-s + (−31.8 − 47.6i)7-s + (48.9 + 118. i)8-s + (38.0 − 15.7i)9-s + (177. − 118. i)10-s + (−33.5 + 168. i)11-s + (−313. − 209. i)12-s + (−34.7 + 34.7i)13-s + (−79.0 − 397. i)14-s + (−127. + 307. i)15-s + 359. i·16-s + (284. − 49.3i)17-s + ⋯ |
| L(s) = 1 | + (1.63 + 0.676i)2-s + (−1.20 + 0.239i)3-s + (1.50 + 1.50i)4-s + (0.669 − 1.00i)5-s + (−2.13 − 0.423i)6-s + (−0.649 − 0.971i)7-s + (0.764 + 1.84i)8-s + (0.470 − 0.194i)9-s + (1.77 − 1.18i)10-s + (−0.277 + 1.39i)11-s + (−2.17 − 1.45i)12-s + (−0.205 + 0.205i)13-s + (−0.403 − 2.02i)14-s + (−0.566 + 1.36i)15-s + 1.40i·16-s + (0.985 − 0.170i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.82039 + 0.738515i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82039 + 0.738515i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-284. + 49.3i)T \) |
| good | 2 | \( 1 + (-6.53 - 2.70i)T + (11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (10.8 - 2.15i)T + (74.8 - 30.9i)T^{2} \) |
| 5 | \( 1 + (-16.7 + 25.0i)T + (-239. - 577. i)T^{2} \) |
| 7 | \( 1 + (31.8 + 47.6i)T + (-918. + 2.21e3i)T^{2} \) |
| 11 | \( 1 + (33.5 - 168. i)T + (-1.35e4 - 5.60e3i)T^{2} \) |
| 13 | \( 1 + (34.7 - 34.7i)T - 2.85e4iT^{2} \) |
| 19 | \( 1 + (117. + 48.4i)T + (9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (327. + 65.1i)T + (2.58e5 + 1.07e5i)T^{2} \) |
| 29 | \( 1 + (24.3 + 16.2i)T + (2.70e5 + 6.53e5i)T^{2} \) |
| 31 | \( 1 + (-200. - 1.00e3i)T + (-8.53e5 + 3.53e5i)T^{2} \) |
| 37 | \( 1 + (-14.9 + 2.96i)T + (1.73e6 - 7.17e5i)T^{2} \) |
| 41 | \( 1 + (867. + 1.29e3i)T + (-1.08e6 + 2.61e6i)T^{2} \) |
| 43 | \( 1 + (-1.97e3 + 817. i)T + (2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (486. - 486. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-2.62e3 - 1.08e3i)T + (5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (2.01e3 + 4.86e3i)T + (-8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (730. - 488. i)T + (5.29e6 - 1.27e7i)T^{2} \) |
| 67 | \( 1 - 4.63e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (6.49e3 - 1.29e3i)T + (2.34e7 - 9.72e6i)T^{2} \) |
| 73 | \( 1 + (1.57e3 - 2.35e3i)T + (-1.08e7 - 2.62e7i)T^{2} \) |
| 79 | \( 1 + (712. - 3.58e3i)T + (-3.59e7 - 1.49e7i)T^{2} \) |
| 83 | \( 1 + (-416. + 1.00e3i)T + (-3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (1.11e4 + 1.11e4i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (727. + 486. i)T + (3.38e7 + 8.17e7i)T^{2} \) |
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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
−17.40727743823333167549480755411, −16.74286733602205633054010347702, −15.84745258509801989610352757520, −14.13964485880718457143258583537, −12.87485572127272320804361830429, −12.12729043008025385070406307930, −10.14862919777715436771184766693, −7.04408597119656956388487276325, −5.58426443473541735171857816923, −4.49011841104287521313384509912,
2.92089721679853394887248299987, 5.74782394408575693934837720779, 6.15082807172924950601790022220, 10.33870350511832124255927471717, 11.43281169136971929105879498513, 12.44685094086332952803518631287, 13.71523219929024857213295918371, 14.95896004979116358613323871012, 16.44003782822195076706766528121, 18.26293998996203350023457701176
