L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (2.40 + 1.10i)7-s + (−0.499 + 0.866i)9-s + (1.90 + 3.29i)11-s + 4.31·13-s − 0.999·15-s + (0.657 + 1.13i)17-s + (3.81 − 6.60i)19-s + (0.247 + 2.63i)21-s + (3.41 − 5.90i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s − 4.30·29-s + (0.747 + 1.29i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.908 + 0.416i)7-s + (−0.166 + 0.288i)9-s + (0.574 + 0.994i)11-s + 1.19·13-s − 0.258·15-s + (0.159 + 0.276i)17-s + (0.875 − 1.51i)19-s + (0.0539 + 0.574i)21-s + (0.711 − 1.23i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s − 0.799·29-s + (0.134 + 0.232i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.254454416\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.254454416\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.40 - 1.10i)T \) |
good | 11 | \( 1 + (-1.90 - 3.29i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 + (-0.657 - 1.13i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.81 + 6.60i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.41 + 5.90i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.30T + 29T^{2} \) |
| 31 | \( 1 + (-0.747 - 1.29i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.65 - 8.05i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 4.80T + 43T^{2} \) |
| 47 | \( 1 + (1.75 - 3.03i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.06 - 7.04i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.15 - 7.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.31 + 5.74i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.72 + 13.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 + (1.91 + 3.30i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.06 - 1.84i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + (5.65 - 9.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.303696329113422071325918933683, −8.816859120529012965890640933015, −8.043933378631384280096000903242, −7.12278922293906688520179890514, −6.39498501118532769129292762765, −5.16826684359226789435974501316, −4.58877387988715818033048738077, −3.59416034837587283790702894997, −2.59781697443951232340216406709, −1.38793272217856040307081568683,
1.00136674719589625973213668074, 1.72418931001745979105539161450, 3.50691588749174051349789040845, 3.81113829906137293701695373592, 5.32352956897425091413426077296, 5.80400065145272947345363232696, 7.00160505107213256562289742459, 7.68803568141425218135878344120, 8.434796060521803988844245170755, 8.904977014587105658425844795070