L(s) = 1 | + (1.13 − 0.841i)2-s + (0.583 − 1.91i)4-s + (−0.335 − 2.55i)5-s + (−0.254 − 0.949i)7-s + (−0.947 − 2.66i)8-s + (−2.52 − 2.61i)10-s + (0.837 − 0.642i)11-s + (−1.15 + 1.50i)13-s + (−1.08 − 0.864i)14-s + (−3.31 − 2.23i)16-s − 4.36·17-s + (0.0481 − 0.0199i)19-s + (−5.07 − 0.845i)20-s + (0.410 − 1.43i)22-s + (−0.476 − 0.127i)23-s + ⋯ |
L(s) = 1 | + (0.803 − 0.595i)2-s + (0.291 − 0.956i)4-s + (−0.150 − 1.14i)5-s + (−0.0961 − 0.358i)7-s + (−0.334 − 0.942i)8-s + (−0.799 − 0.827i)10-s + (0.252 − 0.193i)11-s + (−0.320 + 0.418i)13-s + (−0.290 − 0.231i)14-s + (−0.829 − 0.557i)16-s − 1.05·17-s + (0.0110 − 0.00457i)19-s + (−1.13 − 0.188i)20-s + (0.0876 − 0.305i)22-s + (−0.0994 − 0.0266i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.486857 - 1.99831i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.486857 - 1.99831i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.13 + 0.841i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.335 + 2.55i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (0.254 + 0.949i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.837 + 0.642i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.15 - 1.50i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 19 | \( 1 + (-0.0481 + 0.0199i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (0.476 + 0.127i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-8.61 - 1.13i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-4.82 + 2.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.37 - 8.14i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.939 + 3.50i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (6.85 + 8.92i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (2.36 + 1.36i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.72 - 6.57i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (5.89 - 0.776i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.240 + 1.83i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-9.22 + 12.0i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-10.6 - 10.6i)T + 71iT^{2} \) |
| 73 | \( 1 + (-6.06 + 6.06i)T - 73iT^{2} \) |
| 79 | \( 1 + (-7.26 + 12.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.23 + 0.952i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-3.67 + 3.67i)T - 89iT^{2} \) |
| 97 | \( 1 + (-2.21 + 3.82i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.957260172558550358062229016938, −9.035765310298872103436202462602, −8.342444387821528106808029920251, −6.94862646144178352160558991575, −6.23986630772640019574579625517, −4.92554534706843186462478511811, −4.56551906456716557837849906440, −3.46003673133224465130054546553, −2.08335566043395590689844204017, −0.75746752049685134087545498672,
2.40781972628515063812154737242, 3.16323089995316458906379139519, 4.30475692721350391928582188284, 5.25079549879371557509589241600, 6.52517003266736952403568471584, 6.68674719210094640754178079518, 7.81157138585947592350899153329, 8.567081625990706112367971845341, 9.702508393440503237529495332169, 10.72749476591656864707269685045