L(s) = 1 | − 2.64i·2-s + (−1 + 2.82i)3-s − 3.00·4-s + 2.82i·5-s + (7.48 + 2.64i)6-s − 7.48·7-s − 2.64i·8-s + (−7.00 − 5.65i)9-s + 7.48·10-s + (3.00 − 8.48i)12-s + 22.4·13-s + 19.7i·14-s + (−8.00 − 2.82i)15-s − 18.9·16-s − 21.1i·17-s + (−14.9 + 18.5i)18-s + ⋯ |
L(s) = 1 | − 1.32i·2-s + (−0.333 + 0.942i)3-s − 0.750·4-s + 0.565i·5-s + (1.24 + 0.440i)6-s − 1.06·7-s − 0.330i·8-s + (−0.777 − 0.628i)9-s + 0.748·10-s + (0.250 − 0.707i)12-s + 1.72·13-s + 1.41i·14-s + (−0.533 − 0.188i)15-s − 1.18·16-s − 1.24i·17-s + (−0.831 + 1.02i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.698451 - 0.987760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.698451 - 0.987760i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1 - 2.82i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.64iT - 4T^{2} \) |
| 5 | \( 1 - 2.82iT - 25T^{2} \) |
| 7 | \( 1 + 7.48T + 49T^{2} \) |
| 13 | \( 1 - 22.4T + 169T^{2} \) |
| 17 | \( 1 + 21.1iT - 289T^{2} \) |
| 19 | \( 1 - 14.9T + 361T^{2} \) |
| 23 | \( 1 + 31.1iT - 529T^{2} \) |
| 29 | \( 1 + 21.1iT - 841T^{2} \) |
| 31 | \( 1 - 30T + 961T^{2} \) |
| 37 | \( 1 + 10T + 1.36e3T^{2} \) |
| 41 | \( 1 + 42.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 36.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 42.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 97.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 42T + 4.48e3T^{2} \) |
| 71 | \( 1 - 65.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 74.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 22.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 21.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 62.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 74T + 9.40e3T^{2} \) |
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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
−10.88733166681759000455739103118, −10.23901363043048731644885255830, −9.511584420427842835655536365645, −8.690600173452106248225687545348, −6.83737207397697500399098337812, −6.05723558002280440820479234604, −4.51870337630517113599828259052, −3.40926173386547506982743245182, −2.83730532132860287924391173826, −0.63650292951615646684212584155,
1.33801316401957136952610428961, 3.37266882361904690293500604173, 5.11849252463177757637651871015, 6.12873561599567231470033137604, 6.47338713524486528421817897006, 7.61853625544454035444026268708, 8.415983371986487449779387551532, 9.168034924062562742060587603924, 10.65646997521070812744295205351, 11.62490655802646324267738186163