L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.22 + 2.12i)5-s + (−1 − 2.44i)7-s − 0.999i·8-s + (2.12 − 1.22i)10-s + (−3.67 + 2.12i)11-s + 0.717i·13-s + (−0.358 + 2.62i)14-s + (−0.5 + 0.866i)16-s + (−1.22 − 2.12i)17-s + (−4.24 − 2.44i)19-s − 2.44·20-s + 4.24·22-s + (−5.19 − 3i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.547 + 0.948i)5-s + (−0.377 − 0.925i)7-s − 0.353i·8-s + (0.670 − 0.387i)10-s + (−1.10 + 0.639i)11-s + 0.198i·13-s + (−0.0958 + 0.700i)14-s + (−0.125 + 0.216i)16-s + (−0.297 − 0.514i)17-s + (−0.973 − 0.561i)19-s − 0.547·20-s + 0.904·22-s + (−1.08 − 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00813458 + 0.0615616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00813458 + 0.0615616i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 + 2.44i)T \) |
good | 5 | \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.717iT - 13T^{2} \) |
| 17 | \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.24 + 2.44i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.75iT - 29T^{2} \) |
| 31 | \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.62 - 4.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + (6.42 - 11.1i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-12.5 + 7.24i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 2.12i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.62 - 2.09i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.74 + 11.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (4.75 - 2.74i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.378 - 0.655i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + (1.52 - 2.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.16iT - 97T^{2} \) |
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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
−11.49753389456938251978474762594, −10.60636138790191862714879471256, −10.31520711222730642647942551666, −9.144650183384299472816740382809, −7.911696544191727793943383774841, −7.24214260996189226927985151562, −6.47836222355563896891884640529, −4.65677218590670936953521271234, −3.49926994953112822885726444830, −2.32297116541641727875467440390,
0.04669036181114440510663954777, 2.17357927810151831613887630906, 3.87550330384995738353840016764, 5.37576144070655509255839838125, 5.96211246255090869008930499851, 7.43353055838662616225410082281, 8.445953345041308220205447973493, 8.720876903590345877199010319592, 9.929294215237202958178362670860, 10.82453242463637781672143041430