L(s) = 1 | + (2.22 + 2.22i)2-s + (−1.83 + 0.762i)3-s + 1.94i·4-s + (−1.91 − 4.62i)5-s + (−5.80 − 2.40i)6-s + (1.06 − 2.57i)7-s + (13.5 − 13.5i)8-s + (−16.2 + 16.2i)9-s + (6.04 − 14.5i)10-s + (−25.1 − 10.4i)11-s + (−1.47 − 3.57i)12-s + 59.7i·13-s + (8.11 − 3.36i)14-s + (7.05 + 7.05i)15-s + 75.7·16-s + (70.0 + 0.790i)17-s + ⋯ |
L(s) = 1 | + (0.788 + 0.788i)2-s + (−0.354 + 0.146i)3-s + 0.242i·4-s + (−0.171 − 0.413i)5-s + (−0.394 − 0.163i)6-s + (0.0575 − 0.138i)7-s + (0.596 − 0.596i)8-s + (−0.603 + 0.603i)9-s + (0.191 − 0.461i)10-s + (−0.689 − 0.285i)11-s + (−0.0355 − 0.0859i)12-s + 1.27i·13-s + (0.154 − 0.0641i)14-s + (0.121 + 0.121i)15-s + 1.18·16-s + (0.999 + 0.0112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.20128 + 0.457375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20128 + 0.457375i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-70.0 - 0.790i)T \) |
good | 2 | \( 1 + (-2.22 - 2.22i)T + 8iT^{2} \) |
| 3 | \( 1 + (1.83 - 0.762i)T + (19.0 - 19.0i)T^{2} \) |
| 5 | \( 1 + (1.91 + 4.62i)T + (-88.3 + 88.3i)T^{2} \) |
| 7 | \( 1 + (-1.06 + 2.57i)T + (-242. - 242. i)T^{2} \) |
| 11 | \( 1 + (25.1 + 10.4i)T + (941. + 941. i)T^{2} \) |
| 13 | \( 1 - 59.7iT - 2.19e3T^{2} \) |
| 19 | \( 1 + (-23.5 - 23.5i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + (194. + 80.7i)T + (8.60e3 + 8.60e3i)T^{2} \) |
| 29 | \( 1 + (-7.67 - 18.5i)T + (-1.72e4 + 1.72e4i)T^{2} \) |
| 31 | \( 1 + (-123. + 51.1i)T + (2.10e4 - 2.10e4i)T^{2} \) |
| 37 | \( 1 + (-141. + 58.4i)T + (3.58e4 - 3.58e4i)T^{2} \) |
| 41 | \( 1 + (100. - 241. i)T + (-4.87e4 - 4.87e4i)T^{2} \) |
| 43 | \( 1 + (224. - 224. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 329. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (219. + 219. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-38.7 + 38.7i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (313. - 756. i)T + (-1.60e5 - 1.60e5i)T^{2} \) |
| 67 | \( 1 + 731.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-581. + 240. i)T + (2.53e5 - 2.53e5i)T^{2} \) |
| 73 | \( 1 + (189. + 458. i)T + (-2.75e5 + 2.75e5i)T^{2} \) |
| 79 | \( 1 + (-83.1 - 34.4i)T + (3.48e5 + 3.48e5i)T^{2} \) |
| 83 | \( 1 + (257. + 257. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 192. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-516. - 1.24e3i)T + (-6.45e5 + 6.45e5i)T^{2} \) |
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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
−18.54431866019366594724207960837, −16.54562015820955345571297735625, −16.25389409081170567016829057354, −14.52013570412367007245665055242, −13.60731216615582276159608912654, −11.96296532257099424455524573791, −10.24603249627244519951376683375, −8.002828821243005778886454333504, −6.07273228724659865446232937980, −4.62884197489022035717104144423,
3.18412007922491424740425045963, 5.50055939819920528126720465314, 7.86632578310231062624988511328, 10.33274802455025213020382053074, 11.67956941135488594805263782240, 12.61957155695915369091982697148, 14.03783715951160659567550228254, 15.40821886289656431463093495059, 17.20851908058689660332698231623, 18.31407672525728572699326406147