| L(s) = 1 | + (−1.31 + 0.510i)2-s + (−1.38 + 1.04i)3-s + (1.47 − 1.34i)4-s + (−1.62 − 1.53i)5-s + (1.28 − 2.08i)6-s + 1.57i·7-s + (−1.26 + 2.53i)8-s + (0.815 − 2.88i)9-s + (2.92 + 1.19i)10-s + (−0.805 − 0.584i)11-s + (−0.634 + 3.40i)12-s + (3.32 − 2.41i)13-s + (−0.802 − 2.07i)14-s + (3.84 + 0.426i)15-s + (0.370 − 3.98i)16-s + (2.67 + 0.869i)17-s + ⋯ |
| L(s) = 1 | + (−0.932 + 0.361i)2-s + (−0.797 + 0.603i)3-s + (0.739 − 0.673i)4-s + (−0.726 − 0.687i)5-s + (0.525 − 0.850i)6-s + 0.593i·7-s + (−0.445 + 0.895i)8-s + (0.271 − 0.962i)9-s + (0.925 + 0.378i)10-s + (−0.242 − 0.176i)11-s + (−0.183 + 0.983i)12-s + (0.921 − 0.669i)13-s + (−0.214 − 0.553i)14-s + (0.993 + 0.110i)15-s + (0.0926 − 0.995i)16-s + (0.649 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.583372 + 0.110117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583372 + 0.110117i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 - 0.510i)T \) |
| 3 | \( 1 + (1.38 - 1.04i)T \) |
| 5 | \( 1 + (1.62 + 1.53i)T \) |
| good | 7 | \( 1 - 1.57iT - 7T^{2} \) |
| 11 | \( 1 + (0.805 + 0.584i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.32 + 2.41i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.67 - 0.869i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.16 - 0.704i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.32 - 2.41i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.85 + 1.90i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.79 - 1.88i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (6.24 - 4.53i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.677 - 0.931i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + (2.73 + 8.42i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-13.3 + 4.33i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.61 + 4.08i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.31 + 3.13i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (14.4 + 4.69i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.39 - 7.37i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.874 + 0.635i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.31 + 0.428i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.02 - 3.14i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (2.65 - 3.64i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.60 - 11.0i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.79270819328895736121773735894, −10.69091323260213108085661936200, −9.986163515877497595620035057896, −8.808728027526492336619259698351, −8.273480644513460345609206747460, −6.98342593055327466622723568992, −5.74903338054386549456790764325, −5.09320985356282582484795002332, −3.41020594715998488407307013345, −0.925171956770063898108211084112,
1.03427915829674827660351874924, 2.87926172650879248955210690061, 4.34574802260621993210191065756, 6.18816994189897871031827332209, 7.06610631616428125899791566892, 7.67288983821420951146577286537, 8.734843892794107888227462093249, 10.21280077279215902754629551757, 10.73963280122667273150362341408, 11.58920090888871292091695243474
