| L(s) = 1 | + (3.43 + 1.42i)2-s + (6.42 − 1.27i)3-s + (−1.53 − 1.53i)4-s + (−8.81 + 13.1i)5-s + (23.8 + 4.74i)6-s + (5.40 + 8.08i)7-s + (−25.8 − 62.4i)8-s + (−35.2 + 14.5i)9-s + (−49.0 + 32.7i)10-s + (14.6 − 73.8i)11-s + (−11.8 − 7.90i)12-s + (−54.9 + 54.9i)13-s + (7.05 + 35.4i)14-s + (−39.7 + 96.0i)15-s − 216. i·16-s + (272. + 96.2i)17-s + ⋯ |
| L(s) = 1 | + (0.858 + 0.355i)2-s + (0.713 − 0.141i)3-s + (−0.0960 − 0.0960i)4-s + (−0.352 + 0.527i)5-s + (0.663 + 0.131i)6-s + (0.110 + 0.165i)7-s + (−0.404 − 0.975i)8-s + (−0.434 + 0.180i)9-s + (−0.490 + 0.327i)10-s + (0.121 − 0.610i)11-s + (−0.0821 − 0.0549i)12-s + (−0.325 + 0.325i)13-s + (0.0360 + 0.181i)14-s + (−0.176 + 0.426i)15-s − 0.845i·16-s + (0.942 + 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.81317 + 0.236871i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81317 + 0.236871i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-272. - 96.2i)T \) |
| good | 2 | \( 1 + (-3.43 - 1.42i)T + (11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (-6.42 + 1.27i)T + (74.8 - 30.9i)T^{2} \) |
| 5 | \( 1 + (8.81 - 13.1i)T + (-239. - 577. i)T^{2} \) |
| 7 | \( 1 + (-5.40 - 8.08i)T + (-918. + 2.21e3i)T^{2} \) |
| 11 | \( 1 + (-14.6 + 73.8i)T + (-1.35e4 - 5.60e3i)T^{2} \) |
| 13 | \( 1 + (54.9 - 54.9i)T - 2.85e4iT^{2} \) |
| 19 | \( 1 + (-439. - 182. i)T + (9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-107. - 21.4i)T + (2.58e5 + 1.07e5i)T^{2} \) |
| 29 | \( 1 + (1.14e3 + 764. i)T + (2.70e5 + 6.53e5i)T^{2} \) |
| 31 | \( 1 + (152. + 767. i)T + (-8.53e5 + 3.53e5i)T^{2} \) |
| 37 | \( 1 + (-2.16e3 + 430. i)T + (1.73e6 - 7.17e5i)T^{2} \) |
| 41 | \( 1 + (950. + 1.42e3i)T + (-1.08e6 + 2.61e6i)T^{2} \) |
| 43 | \( 1 + (595. - 246. i)T + (2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (2.44e3 - 2.44e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (2.29e3 + 949. i)T + (5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-667. - 1.61e3i)T + (-8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (1.27e3 - 853. i)T + (5.29e6 - 1.27e7i)T^{2} \) |
| 67 | \( 1 - 6.01e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (325. - 64.7i)T + (2.34e7 - 9.72e6i)T^{2} \) |
| 73 | \( 1 + (-4.17e3 + 6.25e3i)T + (-1.08e7 - 2.62e7i)T^{2} \) |
| 79 | \( 1 + (1.53e3 - 7.70e3i)T + (-3.59e7 - 1.49e7i)T^{2} \) |
| 83 | \( 1 + (-4.52e3 + 1.09e4i)T + (-3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (-5.84e3 - 5.84e3i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (3.92e3 + 2.62e3i)T + (3.38e7 + 8.17e7i)T^{2} \) |
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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
−18.59064028278188903232807791654, −16.57952667005700083319070183023, −15.00060628503674106609621920619, −14.35821628866723511855788336519, −13.25640928852572938567836156236, −11.53347116373330781417051728418, −9.492331497891382945486337004269, −7.67463014608414555103634179226, −5.69299559879402104262605696446, −3.45141746248805992042351326497,
3.24557705190703145414027580954, 5.01074272687275574066837057825, 7.920339892252553455522990761924, 9.358344067469760311147020394246, 11.62187081541513974990351610352, 12.74351099943732069876252987052, 14.06220927062991430834096897780, 14.98541028759659927374505134450, 16.70980287892808804261650455888, 18.11170363047312476271222791356
