L(s) = 1 | + 2.58i·2-s + (−0.259 + 0.449i)3-s − 4.70·4-s + (−1.39 − 0.806i)5-s + (−1.16 − 0.671i)6-s + (1.06 + 2.42i)7-s − 6.99i·8-s + (1.36 + 2.36i)9-s + (2.08 − 3.61i)10-s + (2.34 + 1.35i)11-s + (1.21 − 2.11i)12-s + (2.36 − 2.71i)13-s + (−6.27 + 2.74i)14-s + (0.723 − 0.417i)15-s + 8.69·16-s − 3.12·17-s + ⋯ |
L(s) = 1 | + 1.83i·2-s + (−0.149 + 0.259i)3-s − 2.35·4-s + (−0.624 − 0.360i)5-s + (−0.474 − 0.273i)6-s + (0.401 + 0.915i)7-s − 2.47i·8-s + (0.455 + 0.788i)9-s + (0.659 − 1.14i)10-s + (0.706 + 0.407i)11-s + (0.351 − 0.609i)12-s + (0.656 − 0.753i)13-s + (−1.67 + 0.734i)14-s + (0.186 − 0.107i)15-s + 2.17·16-s − 0.758·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.102881 + 0.808077i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.102881 + 0.808077i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.06 - 2.42i)T \) |
| 13 | \( 1 + (-2.36 + 2.71i)T \) |
good | 2 | \( 1 - 2.58iT - 2T^{2} \) |
| 3 | \( 1 + (0.259 - 0.449i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.39 + 0.806i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.34 - 1.35i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + (-3.18 + 1.84i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 1.98T + 23T^{2} \) |
| 29 | \( 1 + (-2.68 - 4.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-9.07 + 5.23i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.95iT - 37T^{2} \) |
| 41 | \( 1 + (6.66 - 3.85i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.67 - 2.90i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.913 - 0.527i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.63 + 6.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 11.4iT - 59T^{2} \) |
| 61 | \( 1 + (-1.46 - 2.53i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.7 - 6.79i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.17 - 0.675i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.88 - 4.55i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.69iT - 83T^{2} \) |
| 89 | \( 1 + 1.75iT - 89T^{2} \) |
| 97 | \( 1 + (13.4 + 7.74i)T + (48.5 + 84.0i)T^{2} \) |
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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
−14.97118413477645884851991268147, −13.86199443567006061363207153034, −12.82081517807083239645848257628, −11.49575939854236929926360378239, −9.768032453162149619297677914334, −8.547437872865687811885774012356, −7.908185700752904829056834577693, −6.55829517220568463113799243894, −5.27188710414152442926613317590, −4.34501875962581247912489375981,
1.26612256890437437309622037185, 3.52586268377497243789815218450, 4.33935097469234951997144780227, 6.71485993270863562220455265898, 8.360790708578977239248235477625, 9.603173921960597174735709023613, 10.63864451778223732063958098317, 11.66814975298971300007850876850, 11.98949995808972793448462448610, 13.53429236003749609406744102316