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
−11.58920090888871292091695243474, −10.73963280122667273150362341408, −10.21280077279215902754629551757, −8.734843892794107888227462093249, −7.67288983821420951146577286537, −7.06610631616428125899791566892, −6.18816994189897871031827332209, −4.34574802260621993210191065756, −2.87926172650879248955210690061, −1.03427915829674827660351874924,
0.925171956770063898108211084112, 3.41020594715998488407307013345, 5.09320985356282582484795002332, 5.74903338054386549456790764325, 6.98342593055327466622723568992, 8.273480644513460345609206747460, 8.808728027526492336619259698351, 9.986163515877497595620035057896, 10.69091323260213108085661936200, 11.79270819328895736121773735894