L(s) = 1 | + (0.830 + 1.14i)2-s + (−0.620 + 1.90i)4-s + (1.88 + 3.26i)5-s + (2.23 + 1.41i)7-s + (−2.69 + 0.868i)8-s + (−2.16 + 4.86i)10-s + (−1.47 − 0.849i)11-s − 5.64i·13-s + (0.233 + 3.73i)14-s + (−3.23 − 2.35i)16-s + (2.26 + 1.30i)17-s + (−1.18 − 2.04i)19-s + (−7.36 + 1.55i)20-s + (−0.249 − 2.38i)22-s + (−0.653 − 1.13i)23-s + ⋯ |
L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.310 + 0.950i)4-s + (0.841 + 1.45i)5-s + (0.844 + 0.535i)7-s + (−0.951 + 0.307i)8-s + (−0.685 + 1.53i)10-s + (−0.443 − 0.256i)11-s − 1.56i·13-s + (0.0624 + 0.998i)14-s + (−0.807 − 0.589i)16-s + (0.548 + 0.316i)17-s + (−0.270 − 0.469i)19-s + (−1.64 + 0.347i)20-s + (−0.0532 − 0.509i)22-s + (−0.136 − 0.236i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.976067 + 1.94116i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.976067 + 1.94116i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.830 - 1.14i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.23 - 1.41i)T \) |
good | 5 | \( 1 + (-1.88 - 3.26i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.47 + 0.849i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.64iT - 13T^{2} \) |
| 17 | \( 1 + (-2.26 - 1.30i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.18 + 2.04i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.653 + 1.13i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + (4.75 + 2.74i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.20 + 3.00i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.10iT - 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 + (-3.53 - 6.12i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.488 - 0.846i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.67 + 3.85i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.64 - 4.41i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.69 + 11.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.40T + 71T^{2} \) |
| 73 | \( 1 + (1.09 - 1.88i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-15.0 + 8.71i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 11.8iT - 83T^{2} \) |
| 89 | \( 1 + (9.58 - 5.53i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−11.09198306080785821276778967453, −10.51963717686701652964436921748, −9.408092595928566792295465106017, −8.135143722130151913541485593243, −7.64043722363921907424671561501, −6.39134058321409574987471686041, −5.80542764347804451560668793914, −4.90463547642269319428213983187, −3.25691972011193190388706443184, −2.48450945942643751098702164664,
1.23745228765813933787973656443, 2.10794933893507612184008390924, 4.01939780209275313738956501466, 4.83058684635302761630484686854, 5.45462004696089261933125051011, 6.70629667114953572622010188856, 8.189927070128698633734518858453, 9.068574263709458012766804354461, 9.793869512441875987049458542179, 10.61862646339861217458541676184