L(s) = 1 | + 4.61i·2-s + (3.98 − 3.33i)3-s − 13.2·4-s + 5·5-s + (15.3 + 18.3i)6-s + (0.582 + 18.5i)7-s − 24.3i·8-s + (4.79 − 26.5i)9-s + 23.0i·10-s + 65.3i·11-s + (−52.9 + 44.2i)12-s + 30.9i·13-s + (−85.4 + 2.68i)14-s + (19.9 − 16.6i)15-s + 6.26·16-s + 85.8·17-s + ⋯ |
L(s) = 1 | + 1.63i·2-s + (0.767 − 0.641i)3-s − 1.66·4-s + 0.447·5-s + (1.04 + 1.25i)6-s + (0.0314 + 0.999i)7-s − 1.07i·8-s + (0.177 − 0.984i)9-s + 0.729i·10-s + 1.79i·11-s + (−1.27 + 1.06i)12-s + 0.660i·13-s + (−1.63 + 0.0513i)14-s + (0.343 − 0.286i)15-s + 0.0978·16-s + 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.798848 + 1.78136i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.798848 + 1.78136i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.98 + 3.33i)T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + (-0.582 - 18.5i)T \) |
good | 2 | \( 1 - 4.61iT - 8T^{2} \) |
| 11 | \( 1 - 65.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 30.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 85.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.24iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 193. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 30.7iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 61.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 8.71T + 5.06e4T^{2} \) |
| 41 | \( 1 - 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 208.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 211. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 213.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 673. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 500.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 319. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 555. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 842.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 887.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 480.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 477. iT - 9.12e5T^{2} \) |
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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
−14.16696703692887448082787000470, −12.85173308629053452675369595530, −12.13563361284230154740676987453, −9.780215739422426033740528898592, −8.996538148687383651397991106482, −7.947404892968539926658717999784, −6.98162165864262504409370016214, −6.03907584314391817163755821129, −4.62708142072470423201108552371, −2.22958578263036092183147928922,
1.14218754413496560872644008707, 3.06514914417123583387143302867, 3.75605219489016252068561277139, 5.43291199370875186424446963458, 7.79075727784437794001129855080, 8.994165890167772490095524225733, 10.01273314867334811736798816436, 10.64143740411520388078266720164, 11.53015181254574881245808176773, 13.08404564254985981726207049427