L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (2 − 3.46i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 6·17-s + 2·19-s + (2.5 + 4.33i)25-s + 3.99·26-s + 0.999·28-s + (3 + 5.19i)29-s + (2 − 3.46i)31-s + (0.499 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (0.554 − 0.960i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s + 1.45·17-s + 0.458·19-s + (0.5 + 0.866i)25-s + 0.784·26-s + 0.188·28-s + (0.557 + 0.964i)29-s + (0.359 − 0.622i)31-s + (0.0883 − 0.153i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.981341956\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.981341956\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.896832065968700209378951010560, −8.981262373361538721293537345588, −8.031293774802245118875663112020, −7.48655076274356680955745994755, −6.51934795054915739147487627171, −5.63891753560402551551713279035, −4.95184129186987004969716310671, −3.67027232218535153757676445394, −3.01118174722901517185218105187, −1.06526607354851973657984272848,
1.10167326553198431223830906085, 2.43237795903141877385950354742, 3.45173581884622919931806138410, 4.38023679322460222737079841298, 5.40485379215751779468256095872, 6.20692714034726877009450547907, 7.13726025543389462193913449045, 8.294617437855192102100987822451, 8.974194983815212186385065008631, 9.965363900033757056542797857688