| L(s) = 1 | + (0.415 + 0.909i)3-s + (−1.13 − 1.30i)5-s + (−2.90 − 1.86i)7-s + (−0.654 + 0.755i)9-s + (−0.0221 − 0.154i)11-s + (1.32 − 0.852i)13-s + (0.717 − 1.57i)15-s + (−3.62 + 1.06i)17-s + (−7.74 − 2.27i)19-s + (0.491 − 3.42i)21-s + (−4.52 − 1.59i)23-s + (0.286 − 1.99i)25-s + (−0.959 − 0.281i)27-s + (−1.19 + 0.350i)29-s + (1.05 − 2.30i)31-s + ⋯ |
| L(s) = 1 | + (0.239 + 0.525i)3-s + (−0.505 − 0.583i)5-s + (−1.09 − 0.706i)7-s + (−0.218 + 0.251i)9-s + (−0.00668 − 0.0464i)11-s + (0.367 − 0.236i)13-s + (0.185 − 0.405i)15-s + (−0.878 + 0.258i)17-s + (−1.77 − 0.521i)19-s + (0.107 − 0.746i)21-s + (−0.942 − 0.333i)23-s + (0.0573 − 0.399i)25-s + (−0.184 − 0.0542i)27-s + (−0.221 + 0.0650i)29-s + (0.189 − 0.414i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.210323 - 0.471206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.210323 - 0.471206i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (4.52 + 1.59i)T \) |
| good | 5 | \( 1 + (1.13 + 1.30i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (2.90 + 1.86i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.0221 + 0.154i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.32 + 0.852i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (3.62 - 1.06i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (7.74 + 2.27i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (1.19 - 0.350i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.05 + 2.30i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-6.00 + 6.92i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.15 - 2.49i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.140 + 0.308i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 + (-1.18 - 0.760i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (11.7 - 7.58i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.18 + 9.17i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.550 - 3.82i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.572 - 3.98i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.07 - 0.316i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (1.22 - 0.785i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (10.7 - 12.3i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.49 - 3.26i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-7.09 - 8.18i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.51997482586452060249747604209, −9.547439436201903062293252293793, −8.737404643267910983077021634624, −7.980919084034838519226847803642, −6.76184739408765635179687227902, −5.93600331271828992938378256945, −4.30942119017671232722280176473, −4.03013591222216024587724934718, −2.52490618348672306496471866980, −0.26865397028732720143582825768,
2.14075114113285956697559930252, 3.20658153091919912242861326786, 4.27749339375928507936673132146, 6.01008785271309188176327238930, 6.48746140331446408752348556565, 7.46899965913277890585506342705, 8.498025582713493563667430757602, 9.208347675100779658071537583729, 10.25053344283352841248315819567, 11.17438822475791424863002423096
