| L(s) = 1 | + (0.245 + 1.39i)2-s + (−0.319 + 2.98i)3-s + (−1.87 + 0.684i)4-s + (−8.21 − 1.44i)5-s + (−4.23 + 0.288i)6-s + (6.97 + 0.636i)7-s + (−1.41 − 2.44i)8-s + (−8.79 − 1.90i)9-s − 11.7i·10-s + (−3.18 − 18.0i)11-s + (−1.44 − 5.82i)12-s + (11.7 − 14.0i)13-s + (0.825 + 9.86i)14-s + (6.93 − 24.0i)15-s + (3.06 − 2.57i)16-s + 13.7i·17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.106 + 0.994i)3-s + (−0.469 + 0.171i)4-s + (−1.64 − 0.289i)5-s + (−0.705 + 0.0480i)6-s + (0.995 + 0.0909i)7-s + (−0.176 − 0.306i)8-s + (−0.977 − 0.211i)9-s − 1.17i·10-s + (−0.289 − 1.64i)11-s + (−0.120 − 0.485i)12-s + (0.904 − 1.07i)13-s + (0.0589 + 0.704i)14-s + (0.462 − 1.60i)15-s + (0.191 − 0.160i)16-s + 0.806i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.802103 - 0.123173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.802103 - 0.123173i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.245 - 1.39i)T \) |
| 3 | \( 1 + (0.319 - 2.98i)T \) |
| 7 | \( 1 + (-6.97 - 0.636i)T \) |
| good | 5 | \( 1 + (8.21 + 1.44i)T + (23.4 + 8.55i)T^{2} \) |
| 11 | \( 1 + (3.18 + 18.0i)T + (-113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (-11.7 + 14.0i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 - 13.7iT - 289T^{2} \) |
| 19 | \( 1 - 28.1iT - 361T^{2} \) |
| 23 | \( 1 + (9.53 + 7.99i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (7.42 - 6.22i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (10.9 + 30.1i)T + (-736. + 617. i)T^{2} \) |
| 37 | \( 1 + (12.8 + 22.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-47.1 + 56.2i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (43.1 + 15.7i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-9.88 + 27.1i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (25.4 + 44.1i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-47.8 + 57.0i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-5.80 + 15.9i)T + (-2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (4.23 - 24.0i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-6.53 + 11.3i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-4.96 - 2.86i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (12.3 + 69.8i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (54.5 + 64.9i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 - 110. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-36.3 + 99.9i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−11.04840455456978966505897155751, −10.46375642124985135841225565855, −8.728023793893557616745252015466, −8.274883743758745736881402245509, −7.81837227818520321393713284901, −5.93818830788871692368841875459, −5.31416760161640706541808818228, −3.90836630656272251534892722887, −3.60442415455161429928312676427, −0.39405826057936636479522535348,
1.34693084239610283512988979353, 2.70085018719634880881769210999, 4.21207316052142709837814727363, 4.93363000182188766608605630422, 6.81171846910687204597803250502, 7.43493822506253541839258593917, 8.255421632694658134688761782067, 9.245461693267654225814330300534, 10.80417013634300810594348385192, 11.52302807315551947214227702133
