| L(s) = 1 | + (−0.870 + 2.10i)2-s + (1.40 + 0.936i)3-s + (−0.833 − 0.833i)4-s + (−4.41 + 0.877i)5-s + (−3.18 + 2.13i)6-s + (−9.06 − 1.80i)7-s + (−5.93 + 2.45i)8-s + (−2.35 − 5.68i)9-s + (1.99 − 10.0i)10-s + (4.92 + 7.36i)11-s + (−0.387 − 1.94i)12-s + (15.9 − 15.9i)13-s + (11.6 − 17.4i)14-s + (−7.00 − 2.90i)15-s − 19.3i·16-s + ⋯ |
| L(s) = 1 | + (−0.435 + 1.05i)2-s + (0.467 + 0.312i)3-s + (−0.208 − 0.208i)4-s + (−0.882 + 0.175i)5-s + (−0.531 + 0.355i)6-s + (−1.29 − 0.257i)7-s + (−0.741 + 0.307i)8-s + (−0.261 − 0.632i)9-s + (0.199 − 1.00i)10-s + (0.447 + 0.669i)11-s + (−0.0322 − 0.162i)12-s + (1.22 − 1.22i)13-s + (0.834 − 1.24i)14-s + (−0.467 − 0.193i)15-s − 1.20i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.157120 - 0.104462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.157120 - 0.104462i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (0.870 - 2.10i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-1.40 - 0.936i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (4.41 - 0.877i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (9.06 + 1.80i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-4.92 - 7.36i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-15.9 + 15.9i)T - 169iT^{2} \) |
| 19 | \( 1 + (-2.25 + 5.44i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (20.0 - 13.4i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (5.05 + 25.4i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (10.2 - 15.3i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (44.6 + 29.8i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (15.5 + 3.09i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-3.80 - 9.18i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (54.2 - 54.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-26.6 + 64.4i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (20.4 - 8.48i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (1.25 - 6.28i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 11.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-17.5 - 11.7i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (68.1 - 13.5i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (61.5 + 92.1i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (31.4 + 13.0i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-6.29 - 6.29i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-17.1 - 86.3i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−11.47084427962435131071640048715, −10.10744258177585908152861731905, −9.290317458526411092892710241603, −8.405173885299858184827315451575, −7.55633774850848635571286213159, −6.61665171115679606654829368085, −5.80997023278515115744837098686, −3.79186046444715202899180694866, −3.20186508591216728027757381015, −0.098040766009095304386435140117,
1.69880633066189110979813935614, 3.14544779487127857365679589333, 3.89589706064159344559199218402, 5.97501628794075999995296531556, 6.87769055458966421826734456148, 8.420284853596026017539658847979, 8.862991844500423713904876556779, 9.922786465676147022287263566152, 10.95852121052895730180930010470, 11.68415127078561718608922756092
