L(s) = 1 | − 2.20·2-s + 2.87·4-s + (−1.90 + 3.29i)5-s + (0.741 + 2.53i)7-s − 1.93·8-s + (4.20 − 7.28i)10-s + (2.16 + 3.74i)11-s + (1.43 + 2.49i)13-s + (−1.63 − 5.60i)14-s − 1.48·16-s + (2.01 − 3.48i)17-s + (0.804 + 1.39i)19-s + (−5.47 + 9.48i)20-s + (−4.77 − 8.26i)22-s + (−1.33 + 2.30i)23-s + ⋯ |
L(s) = 1 | − 1.56·2-s + 1.43·4-s + (−0.851 + 1.47i)5-s + (0.280 + 0.959i)7-s − 0.683·8-s + (1.32 − 2.30i)10-s + (0.651 + 1.12i)11-s + (0.398 + 0.690i)13-s + (−0.437 − 1.49i)14-s − 0.370·16-s + (0.488 − 0.845i)17-s + (0.184 + 0.319i)19-s + (−1.22 + 2.12i)20-s + (−1.01 − 1.76i)22-s + (−0.278 + 0.481i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.122495 + 0.503444i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.122495 + 0.503444i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.741 - 2.53i)T \) |
good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 5 | \( 1 + (1.90 - 3.29i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.16 - 3.74i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.43 - 2.49i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.01 + 3.48i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.804 - 1.39i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.33 - 2.30i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.375 + 0.649i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 0.140T + 31T^{2} \) |
| 37 | \( 1 + (-4.14 - 7.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.18 + 8.98i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.133 - 0.231i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 7.93T + 47T^{2} \) |
| 53 | \( 1 + (5.61 - 9.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 0.693T + 59T^{2} \) |
| 61 | \( 1 - 2.10T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 + (-1.78 + 3.09i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + (-3.22 + 5.57i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.128 - 0.222i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.529 - 0.916i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.97550992976930263196826350532, −10.09081767489056496044867610860, −9.402299747120081548498997767447, −8.533798960518710043072864942695, −7.51296475274756407393194617144, −7.11409193228769615980359066954, −6.11814954351020503961742459389, −4.38103904915766974318228134287, −2.96652525120167619703073110451, −1.79646061211357969202039218632,
0.56947689816036117934103958441, 1.31317951163011254404589383403, 3.60067368382277394251181872553, 4.59276173147705706289166782665, 5.97461776253454374713632525072, 7.26987255355744264419054548159, 8.196748228953002309029623332894, 8.393698027007476013567273609193, 9.288989009125575281818645353892, 10.27441416992368054880338240289