L(s) = 1 | + (0.366 + 1.36i)2-s + (0.866 − 1.5i)3-s + (−1.73 + i)4-s + (1 + 0.267i)5-s + (2.36 + 0.633i)6-s + (2.36 + 1.36i)7-s + (−2 − 1.99i)8-s + (−1.5 − 2.59i)9-s + 1.46i·10-s + (1.13 + 4.23i)11-s + 3.46i·12-s + (0.901 − 3.36i)13-s + (−0.999 + 3.73i)14-s + (1.26 − 1.26i)15-s + (1.99 − 3.46i)16-s − 5.73·17-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (0.499 − 0.866i)3-s + (−0.866 + 0.5i)4-s + (0.447 + 0.119i)5-s + (0.965 + 0.258i)6-s + (0.894 + 0.516i)7-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)9-s + 0.462i·10-s + (0.341 + 1.27i)11-s + 0.999i·12-s + (0.250 − 0.933i)13-s + (−0.267 + 0.997i)14-s + (0.327 − 0.327i)15-s + (0.499 − 0.866i)16-s − 1.39·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33263 + 0.518231i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33263 + 0.518231i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
good | 5 | \( 1 + (-1 - 0.267i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-2.36 - 1.36i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.13 - 4.23i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.901 + 3.36i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + (2.36 + 2.36i)T + 19iT^{2} \) |
| 23 | \( 1 + (4.09 - 2.36i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.36 + 0.633i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (0.267 + 0.464i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.73 + 4.73i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.23 - 8.33i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-3.83 + 6.63i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.46 - 7.46i)T - 53iT^{2} \) |
| 59 | \( 1 + (-7.33 - 1.96i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-11.1 + 3i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (1.76 - 6.59i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 2.92iT - 71T^{2} \) |
| 73 | \( 1 + 6.26iT - 73T^{2} \) |
| 79 | \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.36 - 0.366i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 2iT - 89T^{2} \) |
| 97 | \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.32083695089443945788203357704, −12.61637220742050504617444878589, −11.53559623076194766982209710882, −9.756896021186314798326279182382, −8.690942904981951103212933914153, −7.86991057058215315590972851660, −6.81034816487123612174399055683, −5.78395227319685843449626916722, −4.34546274203470849940350431922, −2.26569361244856301475568674860,
2.03927772605864394853899138606, 3.79695875238478591206756505681, 4.65353205644240147684521217007, 6.06638622518396496763135835197, 8.317960413009994173860736631235, 8.949463079538421767894310194195, 10.08277951912110871675076194763, 11.02287539217196225562641903973, 11.57381781142887454534009913713, 13.23596127546621509435153739730