L(s) = 1 | + (1.5 + 0.866i)5-s + (−1.5 + 0.866i)7-s + (1.5 + 2.59i)11-s + (−2.5 + 4.33i)13-s − 6.92i·17-s + 3.46i·19-s + (−4.5 + 7.79i)23-s + (−1 − 1.73i)25-s + (1.5 − 0.866i)29-s + (−4.5 − 2.59i)31-s − 3·35-s − 2·37-s + (4.5 + 2.59i)41-s + (−4.5 + 2.59i)43-s + (−1.5 − 2.59i)47-s + ⋯ |
L(s) = 1 | + (0.670 + 0.387i)5-s + (−0.566 + 0.327i)7-s + (0.452 + 0.783i)11-s + (−0.693 + 1.20i)13-s − 1.68i·17-s + 0.794i·19-s + (−0.938 + 1.62i)23-s + (−0.200 − 0.346i)25-s + (0.278 − 0.160i)29-s + (−0.808 − 0.466i)31-s − 0.507·35-s − 0.328·37-s + (0.702 + 0.405i)41-s + (−0.686 + 0.396i)43-s + (−0.218 − 0.378i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.142691925\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.142691925\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6.92iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 0.866i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.5 + 2.59i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 - 2.59i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.5 + 4.33i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (7.5 - 4.33i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.592472324899106747389898794433, −9.248718834402307276561432462140, −7.86759293095456980613272584971, −7.13208262648442434224892494159, −6.45872946180404446083530780377, −5.64682209067740815230690387265, −4.69797866011886189751019091636, −3.70139740425310828969702479608, −2.53473145856749289354138095208, −1.73917767817959871792773412354,
0.40191272026119507821178445512, 1.81786518851975759053170609093, 3.02695033543572044774826826771, 3.92740572685284767664605056745, 5.01798266307769108217538464752, 5.91177371719511057388272379425, 6.43153746549980052045646729457, 7.46634096739167109058106626830, 8.440258879515500557185043099971, 8.914850832908453786964187646124