L(s) = 1 | + (−0.737 − 2.90i)3-s + (1.57 + 1.57i)5-s + 3.64i·7-s + (−7.91 + 4.29i)9-s + (1.19 + 1.19i)11-s + (14.6 + 14.6i)13-s + (3.41 − 5.74i)15-s + 28.0i·17-s + (−12.5 − 12.5i)19-s + (10.6 − 2.69i)21-s + 29.2·23-s − 20.0i·25-s + (18.3 + 19.8i)27-s + (−19.3 + 19.3i)29-s + 11.6·31-s + ⋯ |
L(s) = 1 | + (−0.245 − 0.969i)3-s + (0.314 + 0.314i)5-s + 0.520i·7-s + (−0.878 + 0.476i)9-s + (0.108 + 0.108i)11-s + (1.12 + 1.12i)13-s + (0.227 − 0.382i)15-s + 1.65i·17-s + (−0.662 − 0.662i)19-s + (0.504 − 0.128i)21-s + 1.27·23-s − 0.801i·25-s + (0.678 + 0.734i)27-s + (−0.667 + 0.667i)29-s + 0.375·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.50884 + 0.328868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50884 + 0.328868i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.737 + 2.90i)T \) |
good | 5 | \( 1 + (-1.57 - 1.57i)T + 25iT^{2} \) |
| 7 | \( 1 - 3.64iT - 49T^{2} \) |
| 11 | \( 1 + (-1.19 - 1.19i)T + 121iT^{2} \) |
| 13 | \( 1 + (-14.6 - 14.6i)T + 169iT^{2} \) |
| 17 | \( 1 - 28.0iT - 289T^{2} \) |
| 19 | \( 1 + (12.5 + 12.5i)T + 361iT^{2} \) |
| 23 | \( 1 - 29.2T + 529T^{2} \) |
| 29 | \( 1 + (19.3 - 19.3i)T - 841iT^{2} \) |
| 31 | \( 1 - 11.6T + 961T^{2} \) |
| 37 | \( 1 + (0.771 - 0.771i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 25.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-40.5 + 40.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 50.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-46.2 - 46.2i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-22.7 - 22.7i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (12.7 + 12.7i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (10.6 + 10.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 122.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 15.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 51.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (37.8 - 37.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 5.45T + 7.92e3T^{2} \) |
| 97 | \( 1 + 81.1T + 9.40e3T^{2} \) |
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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
−11.12880527682592982096512853231, −10.62753003553220604954658558882, −8.994854211884107817750082973060, −8.574261495140484286896261469853, −7.23456373187493636122188512431, −6.38246399794921478451144298736, −5.75610095848631359515124083686, −4.18857128215640756086530404426, −2.57425309045472827535512621903, −1.42668546379484756089983495971,
0.78452393529266808689949882403, 2.99203110125751283456903362659, 4.07017583473527598462663020204, 5.21450538690892794372967908910, 5.97383070141483690800940184559, 7.31256043946777396423158628899, 8.532613679463037723148350846474, 9.308748593572388468149302114571, 10.19531231260327681567587365547, 10.97206219588766732891320137369