| L(s) = 1 | + (0.951 + 0.309i)2-s + (0.578 − 0.795i)3-s + (0.809 + 0.587i)4-s + (1.99 − 1.00i)5-s + (0.795 − 0.578i)6-s + i·7-s + (0.587 + 0.809i)8-s + (0.628 + 1.93i)9-s + (2.21 − 0.334i)10-s + (1.14 − 3.52i)11-s + (0.935 − 0.303i)12-s + (−6.30 + 2.04i)13-s + (−0.309 + 0.951i)14-s + (0.358 − 2.16i)15-s + (0.309 + 0.951i)16-s + (−1.84 − 2.54i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (0.333 − 0.459i)3-s + (0.404 + 0.293i)4-s + (0.894 − 0.447i)5-s + (0.324 − 0.235i)6-s + 0.377i·7-s + (0.207 + 0.286i)8-s + (0.209 + 0.644i)9-s + (0.699 − 0.105i)10-s + (0.344 − 1.06i)11-s + (0.269 − 0.0877i)12-s + (−1.74 + 0.568i)13-s + (−0.0825 + 0.254i)14-s + (0.0925 − 0.560i)15-s + (0.0772 + 0.237i)16-s + (−0.448 − 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.37117 - 0.0905301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37117 - 0.0905301i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (-1.99 + 1.00i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (-0.578 + 0.795i)T + (-0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (-1.14 + 3.52i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (6.30 - 2.04i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.84 + 2.54i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.50 + 1.09i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.99 - 1.29i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (1.14 + 0.828i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (7.02 - 5.10i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (4.73 - 1.53i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (2.79 + 8.59i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.81iT - 43T^{2} \) |
| 47 | \( 1 + (3.21 - 4.41i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.27 - 3.12i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.30 - 7.10i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.36 + 13.4i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.688 - 0.947i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (6.63 + 4.82i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (11.3 + 3.69i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.20 - 6.68i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (1.60 + 2.20i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (1.91 - 5.89i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.07 - 1.47i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.72362720317590265931653939398, −10.67082464829798458394772319088, −9.418124549099532604939003975489, −8.737928197145194999780112635374, −7.46043410875627672002413445045, −6.69896160504686083022422011588, −5.39674527838785843741007162044, −4.80604161109610301720480563442, −2.96092082573490382847225094290, −1.88907302809611767679687240860,
1.99033830223470108161804483761, 3.22741701167880125880948504095, 4.43195167240417053544013754754, 5.41541026019553329085589752818, 6.72458601965275116529830873118, 7.35416105726557667741939528431, 9.078076495195913409780933118912, 9.953093240542584729708047775625, 10.27990047251049723889676404047, 11.56561861708768648992850235733
