| L(s) = 1 | + (−22.4 − 15.0i)3-s + 122.·5-s + 206.·7-s + (275. + 674. i)9-s + 142.·11-s + 3.49e3i·13-s + (−2.75e3 − 1.84e3i)15-s + 3.38e3i·17-s − 755. i·19-s + (−4.63e3 − 3.10e3i)21-s + 1.69e4i·23-s − 561.·25-s + (3.96e3 − 1.92e4i)27-s − 4.11e4·29-s − 2.29e4·31-s + ⋯ |
| L(s) = 1 | + (−0.830 − 0.557i)3-s + 0.981·5-s + 0.602·7-s + (0.378 + 0.925i)9-s + 0.107·11-s + 1.59i·13-s + (−0.815 − 0.547i)15-s + 0.688i·17-s − 0.110i·19-s + (−0.500 − 0.335i)21-s + 1.38i·23-s − 0.0359·25-s + (0.201 − 0.979i)27-s − 1.68·29-s − 0.769·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.6370044995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6370044995\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (22.4 + 15.0i)T \) |
| good | 5 | \( 1 - 122.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 206.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 142.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 3.49e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 3.38e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 755. iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.69e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 4.11e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.29e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.68e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 4.04e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.28e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 6.06e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 5.29e3T + 2.21e10T^{2} \) |
| 59 | \( 1 - 9.73e4T + 4.21e10T^{2} \) |
| 61 | \( 1 + 2.13e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 3.85e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.56e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.41e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 7.22e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 8.04e3T + 3.26e11T^{2} \) |
| 89 | \( 1 + 4.10e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.52e6T + 8.32e11T^{2} \) |
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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
−10.93928190998868835510462292802, −9.790683139369840544409233093644, −9.011256522007495868864334194052, −7.70342881267217184240062949521, −6.84228919461190063526149278788, −5.89871843526035600103835274253, −5.18218956522141210284917892891, −3.94508009503308979327078664826, −1.85714352207533761366641014560, −1.66190650762518891453499278558,
0.15042249454059650353842047774, 1.35392587858220824903084327605, 2.77690179936766228526139723162, 4.19434127329613658793436304816, 5.36690053443844621832497746165, 5.75805457766217923659788562562, 6.96448304578157436697253590941, 8.150804644483838652063069944474, 9.332151197910673157050833640846, 10.04135428534555462037710200502
