L(s) = 1 | + (−0.5 + 0.866i)3-s + (1.33 − 2.31i)5-s + 3.93i·7-s + (−0.499 − 0.866i)9-s + 2.20i·11-s + (−3.60 + 2.07i)13-s + (1.33 + 2.31i)15-s + (−0.571 + 0.990i)17-s + (−4.24 + 0.990i)19-s + (−3.40 − 1.96i)21-s + (3.19 − 1.84i)23-s + (−1.07 − 1.85i)25-s + 0.999·27-s + (−2.10 + 1.21i)29-s + 3.67·31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.597 − 1.03i)5-s + 1.48i·7-s + (−0.166 − 0.288i)9-s + 0.664i·11-s + (−0.998 + 0.576i)13-s + (0.345 + 0.597i)15-s + (−0.138 + 0.240i)17-s + (−0.973 + 0.227i)19-s + (−0.743 − 0.429i)21-s + (0.665 − 0.384i)23-s + (−0.214 − 0.371i)25-s + 0.192·27-s + (−0.390 + 0.225i)29-s + 0.659·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.672913 + 0.938983i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.672913 + 0.938983i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (4.24 - 0.990i)T \) |
good | 5 | \( 1 + (-1.33 + 2.31i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 3.93iT - 7T^{2} \) |
| 11 | \( 1 - 2.20iT - 11T^{2} \) |
| 13 | \( 1 + (3.60 - 2.07i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.571 - 0.990i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.19 + 1.84i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.10 - 1.21i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.67T + 31T^{2} \) |
| 37 | \( 1 - 10.0iT - 37T^{2} \) |
| 41 | \( 1 + (-8.01 - 4.62i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.490 + 0.283i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (10.6 - 6.13i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.09 + 2.36i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.33 - 2.31i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.41 + 11.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 - 3.00i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.24 - 10.8i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.35 + 2.34i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.50 - 6.07i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.98iT - 83T^{2} \) |
| 89 | \( 1 + (-12.9 + 7.46i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.91 - 1.68i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.05733931430696775926678065007, −9.466500476537429235446644543468, −8.883145746106880894646508550545, −8.121789367919621494001434515661, −6.68020446043166065849219649282, −5.88923251780886233115620326634, −4.92971500856206461736936925461, −4.54807669791993560211434758134, −2.76438800254682911411037354393, −1.72725514340853779718132374066,
0.54913175304252410757909833278, 2.21617769611382779068480665309, 3.27092515114420637748436822852, 4.47641183444139684971238358788, 5.66760738144838272660418886829, 6.55464613808704802647735445788, 7.22558903476436411927378056845, 7.80437585935607525710176023125, 9.099024735228211403654947566255, 10.15031033485828747100578916440