| L(s) = 1 | + (0.216 + 0.0897i)2-s + (0.273 + 1.37i)3-s + (−2.78 − 2.78i)4-s + (−0.286 − 0.191i)5-s + (−0.0641 + 0.322i)6-s + (−5.96 + 3.98i)7-s + (−0.713 − 1.72i)8-s + (6.49 − 2.69i)9-s + (−0.0449 − 0.0672i)10-s + (12.8 + 2.56i)11-s + (3.07 − 4.59i)12-s + (−5.20 + 5.20i)13-s + (−1.65 + 0.328i)14-s + (0.184 − 0.446i)15-s + 15.3i·16-s + (0.525 − 16.9i)17-s + ⋯ |
| L(s) = 1 | + (0.108 + 0.0448i)2-s + (0.0911 + 0.458i)3-s + (−0.697 − 0.697i)4-s + (−0.0573 − 0.0383i)5-s + (−0.0106 + 0.0537i)6-s + (−0.852 + 0.569i)7-s + (−0.0891 − 0.215i)8-s + (0.722 − 0.299i)9-s + (−0.00449 − 0.00672i)10-s + (1.17 + 0.233i)11-s + (0.256 − 0.383i)12-s + (−0.400 + 0.400i)13-s + (−0.118 + 0.0234i)14-s + (0.0123 − 0.0297i)15-s + 0.958i·16-s + (0.0308 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.781660 + 0.0425657i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.781660 + 0.0425657i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-0.525 + 16.9i)T \) |
| good | 2 | \( 1 + (-0.216 - 0.0897i)T + (2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (-0.273 - 1.37i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (0.286 + 0.191i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (5.96 - 3.98i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (-12.8 - 2.56i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (5.20 - 5.20i)T - 169iT^{2} \) |
| 19 | \( 1 + (22.2 + 9.23i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (6.25 - 31.4i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-21.4 + 32.1i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (-10.5 + 2.09i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-2.37 - 11.9i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (30.0 - 20.1i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-29.2 + 12.1i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (28.8 - 28.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-53.8 - 22.2i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (1.99 + 4.80i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-25.4 - 38.1i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + 59.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (7.79 + 39.1i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (9.83 + 6.57i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (114. + 22.7i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-39.1 + 94.6i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-103. - 103. i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (73.5 - 110. i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−18.94876405558001312684574482962, −17.56163034795585890682893436721, −15.89669161830879471363214185552, −14.92823623187364543917111525687, −13.55418857081328074242134463210, −12.04017273058309333389060148478, −9.903502679536666890974715260870, −9.180973547802950940891479458000, −6.44030584999493844216701907112, −4.34969685742836502956987770253,
3.97512048985636004811259134783, 6.81275856473775933151646775250, 8.477671329738997284771710937247, 10.20040006673002244015773430976, 12.38487545985949393643219399370, 13.13359102357403541357083965265, 14.49436166989012611756006237732, 16.44892201618554943347941824408, 17.35440952993404957991207469085, 18.83058559399559899126261192989
