L(s) = 1 | − 2.49·2-s + 1.73·3-s + 2.21·4-s − 2.66i·5-s − 4.31·6-s + 2.64i·7-s + 4.44·8-s + 2.99·9-s + 6.65i·10-s − 5.14i·11-s + 3.84·12-s + 6.24·13-s − 6.59i·14-s − 4.62i·15-s − 19.9·16-s + 11.3i·17-s + ⋯ |
L(s) = 1 | − 1.24·2-s + 0.577·3-s + 0.554·4-s − 0.533i·5-s − 0.719·6-s + 0.377i·7-s + 0.555·8-s + 0.333·9-s + 0.665i·10-s − 0.468i·11-s + 0.320·12-s + 0.480·13-s − 0.471i·14-s − 0.308i·15-s − 1.24·16-s + 0.670i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.088952732\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.088952732\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73T \) |
| 7 | \( 1 - 2.64iT \) |
| 23 | \( 1 + (-10.8 + 20.2i)T \) |
good | 2 | \( 1 + 2.49T + 4T^{2} \) |
| 5 | \( 1 + 2.66iT - 25T^{2} \) |
| 11 | \( 1 + 5.14iT - 121T^{2} \) |
| 13 | \( 1 - 6.24T + 169T^{2} \) |
| 17 | \( 1 - 11.3iT - 289T^{2} \) |
| 19 | \( 1 - 9.41iT - 361T^{2} \) |
| 29 | \( 1 - 27.6T + 841T^{2} \) |
| 31 | \( 1 + 52.6T + 961T^{2} \) |
| 37 | \( 1 + 49.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 15.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 68.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 95.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 78.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 57.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 93.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 117.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 97.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 140. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 74.2iT - 9.40e3T^{2} \) |
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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
−10.53722989141968481047212612983, −9.539534885308814477997617160589, −8.755947322854697131065213858855, −8.414501996212224119832323321503, −7.45057665327548001021515780057, −6.28392273953778039610514458788, −4.95420555625533164228933141829, −3.70087667966837774751623665233, −2.12949436271266369427463492694, −0.838352189578757602749686673596,
1.04516185886174634143458840248, 2.47388987781180774084876498942, 3.81802783249516185303637074307, 5.10061402949150562729305637640, 6.87013018133030002460323372794, 7.26716985195226075405268329058, 8.292148510698305463934087188630, 9.104903788670086371740737711250, 9.767503835219933173800544035058, 10.65763696912301676782841224100