L(s) = 1 | + (−0.817 − 1.15i)2-s + (0.540 − 0.841i)3-s + (−0.663 + 1.88i)4-s + (−2.63 − 1.20i)5-s + (−1.41 + 0.0637i)6-s + (3.10 + 0.910i)7-s + (2.71 − 0.776i)8-s + (−0.415 − 0.909i)9-s + (0.766 + 4.03i)10-s + (−1.24 − 1.08i)11-s + (1.22 + 1.57i)12-s + (−0.884 − 3.01i)13-s + (−1.48 − 4.32i)14-s + (−2.44 + 1.56i)15-s + (−3.11 − 2.50i)16-s + (−0.685 − 4.76i)17-s + ⋯ |
L(s) = 1 | + (−0.578 − 0.816i)2-s + (0.312 − 0.485i)3-s + (−0.331 + 0.943i)4-s + (−1.18 − 0.539i)5-s + (−0.576 + 0.0260i)6-s + (1.17 + 0.344i)7-s + (0.961 − 0.274i)8-s + (−0.138 − 0.303i)9-s + (0.242 + 1.27i)10-s + (−0.376 − 0.326i)11-s + (0.354 + 0.455i)12-s + (−0.245 − 0.835i)13-s + (−0.396 − 1.15i)14-s + (−0.630 + 0.405i)15-s + (−0.779 − 0.625i)16-s + (−0.166 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0298844 + 0.664277i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0298844 + 0.664277i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.817 + 1.15i)T \) |
| 3 | \( 1 + (-0.540 + 0.841i)T \) |
| 23 | \( 1 + (4.70 - 0.912i)T \) |
good | 5 | \( 1 + (2.63 + 1.20i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-3.10 - 0.910i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (1.24 + 1.08i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.884 + 3.01i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.685 + 4.76i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.389 + 0.0560i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (4.10 - 0.589i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (4.60 - 2.95i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (3.39 - 1.54i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.52 + 3.33i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.82 + 9.06i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 8.73T + 47T^{2} \) |
| 53 | \( 1 + (3.74 - 12.7i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (1.25 + 4.26i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (3.95 + 6.15i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.33 + 1.15i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (8.80 + 10.1i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.408 + 2.84i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (1.26 - 0.370i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-7.35 + 3.35i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (0.761 + 0.489i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (4.43 - 9.70i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.59053604906505692788257968510, −9.186886564833761748181055476780, −8.585203372313536944816874725540, −7.68155129297204675318221983533, −7.49519253047118539809699351865, −5.41575053592694247232855365649, −4.40552333529125316304411228344, −3.26903968820319174119592336698, −1.99223173143837623522658471298, −0.44437378733633520875561094672,
1.94324849755532595821250350610, 3.98884336645621089218911168194, 4.50897211687574254197979271666, 5.80854769849410736365441484195, 7.10861017145113428642505735348, 7.78183324770668809853547619515, 8.305701730315449984671337901282, 9.324800614840316110352823773884, 10.38129499604789617255967660062, 11.02005365619892310456979022798