| L(s) = 1 | + (0.471 + 1.29i)2-s + (−1.37 − 1.05i)3-s + (0.0766 − 0.0643i)4-s + (0.459 + 2.60i)5-s + (0.710 − 2.27i)6-s + (1.90 − 1.83i)7-s + (2.50 + 1.44i)8-s + (0.794 + 2.89i)9-s + (−3.15 + 1.82i)10-s + (−0.681 − 0.120i)11-s + (−0.173 + 0.00811i)12-s + (−0.171 + 0.472i)13-s + (3.27 + 1.61i)14-s + (2.10 − 4.06i)15-s + (−0.658 + 3.73i)16-s + (2.42 + 4.19i)17-s + ⋯ |
| L(s) = 1 | + (0.333 + 0.915i)2-s + (−0.795 − 0.606i)3-s + (0.0383 − 0.0321i)4-s + (0.205 + 1.16i)5-s + (0.290 − 0.930i)6-s + (0.721 − 0.692i)7-s + (0.886 + 0.511i)8-s + (0.264 + 0.964i)9-s + (−0.998 + 0.576i)10-s + (−0.205 − 0.0362i)11-s + (−0.0499 + 0.00234i)12-s + (−0.0476 + 0.130i)13-s + (0.874 + 0.430i)14-s + (0.542 − 1.05i)15-s + (−0.164 + 0.933i)16-s + (0.587 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15631 + 0.647612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15631 + 0.647612i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.37 + 1.05i)T \) |
| 7 | \( 1 + (-1.90 + 1.83i)T \) |
| good | 2 | \( 1 + (-0.471 - 1.29i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.459 - 2.60i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (0.681 + 0.120i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.171 - 0.472i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.42 - 4.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.03 - 0.597i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.74 + 5.65i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.19 + 6.03i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (5.17 + 6.16i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (3.71 + 6.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.25 + 1.18i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.81 - 10.2i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.93 - 2.45i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (1.13 + 6.41i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.36 - 2.81i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (3.24 + 1.17i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.286 + 0.165i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.13 + 1.81i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.71 + 3.53i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (12.8 - 4.69i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-4.84 + 8.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.19 - 0.563i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.91154998163774991354084810302, −11.59223281478882457870100939677, −10.75557994766309321751449112124, −10.26570831527330528996390301105, −7.949839772258037215362409403799, −7.43944132369589661817948543087, −6.38546557659701771391914826712, −5.74383689386350199683601543093, −4.32738840079582397238974713363, −1.98488217511651218881131303379,
1.53490183792937953154354040771, 3.48300301554680462689480146700, 4.96611733145254839111199065916, 5.37286377772622281800517317152, 7.24340215002242424112295467120, 8.682806960905076490077124879647, 9.676312512729973122159380928210, 10.64939354440592470764756047399, 11.77105136886318375775940874324, 12.03285823377709116919916050278
