| L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.0270 − 0.188i)3-s + (0.415 + 0.909i)4-s + (−0.254 − 0.0747i)5-s + (0.0790 − 0.173i)6-s + (0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (2.84 − 0.834i)9-s + (−0.173 − 0.200i)10-s + (2.46 − 1.58i)11-s + (0.160 − 0.102i)12-s + (2.25 + 2.59i)13-s + (0.959 − 0.281i)14-s + (−0.00718 + 0.0499i)15-s + (−0.654 + 0.755i)16-s + (−2.92 + 6.39i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.0156 − 0.108i)3-s + (0.207 + 0.454i)4-s + (−0.113 − 0.0334i)5-s + (0.0322 − 0.0706i)6-s + (0.247 − 0.285i)7-s + (−0.0503 + 0.349i)8-s + (0.947 − 0.278i)9-s + (−0.0549 − 0.0633i)10-s + (0.744 − 0.478i)11-s + (0.0462 − 0.0296i)12-s + (0.624 + 0.720i)13-s + (0.256 − 0.0752i)14-s + (−0.00185 + 0.0128i)15-s + (−0.163 + 0.188i)16-s + (−0.708 + 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.87153 + 0.470075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.87153 + 0.470075i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 7 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (3.69 + 3.05i)T \) |
| good | 3 | \( 1 + (0.0270 + 0.188i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (0.254 + 0.0747i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-2.46 + 1.58i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.25 - 2.59i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.92 - 6.39i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.05 - 2.30i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.23 + 7.07i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.292 + 2.03i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (1.01 - 0.298i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (10.7 + 3.16i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.0614 - 0.427i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 9.70T + 47T^{2} \) |
| 53 | \( 1 + (7.70 - 8.89i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (9.34 + 10.7i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.217 - 1.51i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (9.37 + 6.02i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.54 - 2.27i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.74 - 10.3i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-5.57 - 6.43i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (9.76 - 2.86i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (0.679 + 4.72i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.02 + 0.888i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.91703632130381931923552024369, −10.90488330996322617840252818869, −9.896877049404761878098995317222, −8.637998584388682160062684489160, −7.84791689323287575755126419348, −6.55616822868106499822339489575, −6.08935584987109840920773135355, −4.28751773058572859292419520369, −3.88794073243204474065724626793, −1.76058476555166608671863454009,
1.62324441771338781710222005763, 3.22117371366219995508563692523, 4.45472649858257025821255674278, 5.31577564158251344408616108696, 6.68361314782225086965103430394, 7.53220241712599056078009038854, 8.947640838792884536153509154060, 9.808522707569357822094609459262, 10.77113591454059549739658387392, 11.66668299037614586800822285508
