| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−0.910 + 1.47i)3-s + (0.173 + 0.984i)4-s + (1.95 + 0.710i)5-s + (1.64 − 0.543i)6-s + (0.173 − 0.984i)7-s + (0.500 − 0.866i)8-s + (−1.34 − 2.68i)9-s + (−1.03 − 1.80i)10-s + (2.65 − 0.967i)11-s + (−1.60 − 0.640i)12-s + (2.40 − 2.01i)13-s + (−0.766 + 0.642i)14-s + (−2.82 + 2.23i)15-s + (−0.939 + 0.342i)16-s + (4.06 + 7.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.525 + 0.850i)3-s + (0.0868 + 0.492i)4-s + (0.873 + 0.317i)5-s + (0.671 − 0.221i)6-s + (0.0656 − 0.372i)7-s + (0.176 − 0.306i)8-s + (−0.447 − 0.894i)9-s + (−0.328 − 0.569i)10-s + (0.801 − 0.291i)11-s + (−0.464 − 0.185i)12-s + (0.666 − 0.559i)13-s + (−0.204 + 0.171i)14-s + (−0.729 + 0.575i)15-s + (−0.234 + 0.0855i)16-s + (0.984 + 1.70i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02271 + 0.288488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02271 + 0.288488i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.910 - 1.47i)T \) |
| 7 | \( 1 + (-0.173 + 0.984i)T \) |
| good | 5 | \( 1 + (-1.95 - 0.710i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-2.65 + 0.967i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.40 + 2.01i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.06 - 7.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.84 - 6.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.429 + 2.43i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.27 - 6.10i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.895 + 5.07i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.11 - 1.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.73 + 1.45i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.30 + 0.476i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.84 - 10.4i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.406T + 53T^{2} \) |
| 59 | \( 1 + (-3.88 - 1.41i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.59 + 14.7i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.74 - 5.65i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (6.92 + 11.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.09 - 7.10i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.36 + 7.01i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.617 + 0.518i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.556 + 0.964i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.5 + 5.31i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.07355786998963048085652161763, −10.28841591102183656043243710892, −10.09837978703266118530887159161, −8.843383854777236470274757766734, −8.053078948945796915106485925930, −6.27096147154755027109901080903, −5.96039806857610018632744149205, −4.22571733874453146814188799044, −3.33464581599704656427150605138, −1.44295903509387543128718076270,
1.10573913444310156673548427112, 2.42170341399237778775780928437, 4.78140195094670395318780467367, 5.72961908615980250285955513118, 6.59692641170292820324621890621, 7.31655367277150766437472605210, 8.617377158512178633638190116202, 9.262068165601684075548940140278, 10.22736818528937195677983891303, 11.50601074741287484659846046703
