| 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
−18.11170363047312476271222791356, −16.70980287892808804261650455888, −14.98541028759659927374505134450, −14.06220927062991430834096897780, −12.74351099943732069876252987052, −11.62187081541513974990351610352, −9.358344067469760311147020394246, −7.920339892252553455522990761924, −5.01074272687275574066837057825, −3.24557705190703145414027580954,
3.45141746248805992042351326497, 5.69299559879402104262605696446, 7.67463014608414555103634179226, 9.492331497891382945486337004269, 11.53347116373330781417051728418, 13.25640928852572938567836156236, 14.35821628866723511855788336519, 15.00060628503674106609621920619, 16.57952667005700083319070183023, 18.59064028278188903232807791654
