L(s) = 1 | + (1 − 1.73i)2-s + (−0.999 − 1.73i)4-s + 2·5-s + (0.5 + 2.59i)7-s + (2 − 3.46i)10-s + 2·11-s + (−0.5 + 0.866i)13-s + (5 + 1.73i)14-s + (1.99 − 3.46i)16-s + (−0.5 − 0.866i)19-s + (−1.99 − 3.46i)20-s + (2 − 3.46i)22-s − 25-s + (0.999 + 1.73i)26-s + (4 − 3.46i)28-s + (2 + 3.46i)29-s + ⋯ |
L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + 0.894·5-s + (0.188 + 0.981i)7-s + (0.632 − 1.09i)10-s + 0.603·11-s + (−0.138 + 0.240i)13-s + (1.33 + 0.462i)14-s + (0.499 − 0.866i)16-s + (−0.114 − 0.198i)19-s + (−0.447 − 0.774i)20-s + (0.426 − 0.738i)22-s − 0.200·25-s + (0.196 + 0.339i)26-s + (0.755 − 0.654i)28-s + (0.371 + 0.643i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.06141 - 1.51807i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.06141 - 1.51807i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-8 - 13.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.79798003878440369625481114458, −9.743085392573748199101155708867, −9.269317929588366839494968781945, −8.081369183952875929268387637442, −6.67401132926729745561805558542, −5.65402413059705802940052578498, −4.84646998447341354601751317251, −3.64715336700452918307007466874, −2.45581462611777429379153939795, −1.66496494276734446870788956236,
1.62440466705983268975251950748, 3.56521460178624938428373487826, 4.60396764432397262044816581145, 5.49715026122201957753385216282, 6.39695686068021408051194423395, 7.07624039652029814015629422302, 7.950808660336606371306068858911, 8.995207274493933631245452393988, 10.13300321262547340111441293056, 10.68895164438280465730206340110