L(s) = 1 | + (31.7 + 4.07i)2-s − 190.·3-s + (990. + 258. i)4-s + (101. − 3.12e3i)5-s + (−6.03e3 − 773. i)6-s − 1.40e4·7-s + (3.03e4 + 1.22e4i)8-s − 2.29e4·9-s + (1.59e4 − 9.87e4i)10-s − 1.73e5i·11-s + (−1.88e5 − 4.91e4i)12-s − 6.64e5i·13-s + (−4.46e5 − 5.72e4i)14-s + (−1.93e4 + 5.93e5i)15-s + (9.14e5 + 5.12e5i)16-s − 1.08e6i·17-s + ⋯ |
L(s) = 1 | + (0.991 + 0.127i)2-s − 0.782·3-s + (0.967 + 0.252i)4-s + (0.0325 − 0.999i)5-s + (−0.775 − 0.0995i)6-s − 0.836·7-s + (0.927 + 0.373i)8-s − 0.388·9-s + (0.159 − 0.987i)10-s − 1.07i·11-s + (−0.756 − 0.197i)12-s − 1.78i·13-s + (−0.829 − 0.106i)14-s + (−0.0254 + 0.781i)15-s + (0.872 + 0.488i)16-s − 0.765i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.03364 - 1.38381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03364 - 1.38381i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-31.7 - 4.07i)T \) |
| 5 | \( 1 + (-101. + 3.12e3i)T \) |
good | 3 | \( 1 + 190.T + 5.90e4T^{2} \) |
| 7 | \( 1 + 1.40e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + 1.73e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 6.64e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + 1.08e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 2.14e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 6.85e6T + 4.14e13T^{2} \) |
| 29 | \( 1 - 1.60e7T + 4.20e14T^{2} \) |
| 31 | \( 1 - 3.87e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 1.29e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 4.48e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + 8.66e7T + 2.16e16T^{2} \) |
| 47 | \( 1 - 2.64e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + 2.29e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + 5.54e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 8.72e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 1.53e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 3.28e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.61e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 3.08e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 1.89e9T + 1.55e19T^{2} \) |
| 89 | \( 1 - 8.69e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 5.78e9iT - 7.37e19T^{2} \) |
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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
−15.89238847036307356543816219281, −13.99134660387636992049124874249, −12.79083469567651136831316533320, −11.91640330618895864975527050360, −10.41793988732531663777515758596, −8.212206271929620619037040417874, −6.09337734475947600500741871272, −5.25689121337060948979793777943, −3.24589326078165015875156297742, −0.56485282334765309036061477972,
2.34232654222475261851951282432, 4.19083278101838239405063583215, 6.11869492684587828344870041307, 6.94562660088247695641756703034, 9.934134538040053424540564040713, 11.26093688452911512197382694157, 12.19314588522749907157756616495, 13.73232662506529512727687350397, 14.87543065049747476510171714905, 16.12745897860909251618820904743