| L(s) = 1 | + (−0.316 − 1.97i)2-s + (1.26 + 0.412i)3-s + (−3.79 + 1.25i)4-s − 3.39·5-s + (0.412 − 2.63i)6-s + (−4.74 − 6.52i)7-s + (3.67 + 7.10i)8-s + (−5.84 − 4.24i)9-s + (1.07 + 6.70i)10-s + (−11.3 − 15.6i)11-s + (−5.33 + 0.0196i)12-s + (−1.03 + 3.19i)13-s + (−11.3 + 11.4i)14-s + (−4.30 − 1.40i)15-s + (12.8 − 9.49i)16-s + (16.9 + 12.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.158 − 0.987i)2-s + (0.423 + 0.137i)3-s + (−0.949 + 0.312i)4-s − 0.679·5-s + (0.0687 − 0.439i)6-s + (−0.677 − 0.932i)7-s + (0.458 + 0.888i)8-s + (−0.648 − 0.471i)9-s + (0.107 + 0.670i)10-s + (−1.03 − 1.42i)11-s + (−0.444 + 0.00164i)12-s + (−0.0798 + 0.245i)13-s + (−0.813 + 0.816i)14-s + (−0.287 − 0.0933i)15-s + (0.804 − 0.593i)16-s + (0.997 + 0.724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00561i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.00561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00183584 + 0.654189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00183584 + 0.654189i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.316 + 1.97i)T \) |
| 31 | \( 1 + (-8.73 + 29.7i)T \) |
| good | 3 | \( 1 + (-1.26 - 0.412i)T + (7.28 + 5.29i)T^{2} \) |
| 5 | \( 1 + 3.39T + 25T^{2} \) |
| 7 | \( 1 + (4.74 + 6.52i)T + (-15.1 + 46.6i)T^{2} \) |
| 11 | \( 1 + (11.3 + 15.6i)T + (-37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (1.03 - 3.19i)T + (-136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (-16.9 - 12.3i)T + (89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (-17.3 + 5.62i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (5.79 - 7.97i)T + (-163. - 503. i)T^{2} \) |
| 29 | \( 1 + (-11.5 - 35.6i)T + (-680. + 494. i)T^{2} \) |
| 37 | \( 1 + 2.06T + 1.36e3T^{2} \) |
| 41 | \( 1 + (15.8 + 48.7i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (-75.3 + 24.4i)T + (1.49e3 - 1.08e3i)T^{2} \) |
| 47 | \( 1 + (19.5 + 6.36i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (43.2 + 31.4i)T + (868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (83.3 + 27.0i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 - 59.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (55.3 - 76.2i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-33.5 + 24.4i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (57.0 - 78.5i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-9.23 + 2.99i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + (-47.7 + 34.6i)T + (2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (35.2 - 25.5i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−12.56652339847715461070302636809, −11.51240543377458956910507783764, −10.64165188800935913627152285922, −9.642026505503351110336841413768, −8.475979273944036715315645264433, −7.62115544774277384806980776601, −5.64056820534431898681302373205, −3.77175878135682564691653011802, −3.12923076812328101285035858968, −0.45129669384303260506095587918,
2.90114356693095177812426553642, 4.81998341341185196775765353569, 5.91249299631116962277662059047, 7.54574371942185881266451710740, 7.972198847110868702783884126549, 9.318693038866916523818707672395, 10.14156657349646783367361119216, 11.93543183737595459488415488203, 12.77059659541860692894374009907, 13.89263969047956117388695121694
