L(s) = 1 | + (0.5 − 0.866i)2-s + (0.292 + 0.507i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.585·6-s − 0.999·8-s + (1.32 − 2.30i)9-s + (−0.499 − 0.866i)10-s + (−2.41 − 4.18i)11-s + (0.292 − 0.507i)12-s − 0.828·13-s + 0.585·15-s + (−0.5 + 0.866i)16-s + (2.70 + 4.68i)17-s + (−1.32 − 2.30i)18-s + (1.70 − 2.95i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.169 + 0.292i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.239·6-s − 0.353·8-s + (0.442 − 0.766i)9-s + (−0.158 − 0.273i)10-s + (−0.727 − 1.26i)11-s + (0.0845 − 0.146i)12-s − 0.229·13-s + 0.151·15-s + (−0.125 + 0.216i)16-s + (0.656 + 1.13i)17-s + (−0.313 − 0.542i)18-s + (0.391 − 0.678i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18336 - 1.27258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18336 - 1.27258i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.292 - 0.507i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (2.41 + 4.18i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + (-2.70 - 4.68i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.70 + 2.95i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.41 + 5.91i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + (-1.41 - 2.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.82 - 3.16i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 + (-5.41 + 9.37i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.24 - 9.08i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.70 - 9.88i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.65 - 11.5i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.82 - 8.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + (3.29 + 5.70i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.585 + 1.01i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + (6.36 - 11.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−10.48600277224350097108032493380, −10.19008858846296134917408832075, −8.893655795852875580180685789277, −8.444630176124162426197085404849, −6.91020201690526679778495290001, −5.82452444109223690570197784781, −4.90794096059190083941813958921, −3.74370077305351233896839344449, −2.78099881698057409319702680027, −0.993803227304078733629208062623,
2.01031083241166446899231275834, 3.29374257017834999970556924054, 4.82692287645340763371641627912, 5.40106594110054251686578017732, 6.87230358838467324004666440180, 7.42916779155024807746476905443, 8.097308375099413384463413235554, 9.588457186991777021647955557611, 10.05483588893447218573185654120, 11.26916214430245628482955299629