| L(s) = 1 | + (−1.41 + 0.0951i)2-s + (−0.615 − 1.89i)3-s + (1.98 − 0.268i)4-s + (−0.987 − 2.00i)5-s + (1.04 + 2.61i)6-s − 1.38i·7-s + (−2.77 + 0.567i)8-s + (−0.783 + 0.568i)9-s + (1.58 + 2.73i)10-s + (−3.04 + 4.18i)11-s + (−1.72 − 3.58i)12-s + (2.49 − 1.81i)13-s + (0.131 + 1.94i)14-s + (−3.19 + 3.10i)15-s + (3.85 − 1.06i)16-s + (−3.63 − 1.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 + 0.0672i)2-s + (−0.355 − 1.09i)3-s + (0.990 − 0.134i)4-s + (−0.441 − 0.897i)5-s + (0.428 + 1.06i)6-s − 0.521i·7-s + (−0.979 + 0.200i)8-s + (−0.261 + 0.189i)9-s + (0.500 + 0.865i)10-s + (−0.917 + 1.26i)11-s + (−0.498 − 1.03i)12-s + (0.692 − 0.503i)13-s + (0.0351 + 0.520i)14-s + (−0.824 + 0.801i)15-s + (0.963 − 0.265i)16-s + (−0.880 − 0.286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0845743 - 0.461407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0845743 - 0.461407i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0951i)T \) |
| 5 | \( 1 + (0.987 + 2.00i)T \) |
| good | 3 | \( 1 + (0.615 + 1.89i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 1.38iT - 7T^{2} \) |
| 11 | \( 1 + (3.04 - 4.18i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.49 + 1.81i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.63 + 1.17i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (6.09 + 1.97i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.168 - 0.232i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-8.75 + 2.84i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.38 + 4.27i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.07 - 2.96i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.28 + 2.38i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 7.54T + 43T^{2} \) |
| 47 | \( 1 + (-0.954 + 0.310i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (1.02 + 3.15i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.05 - 1.44i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.30 + 1.80i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.552 - 1.69i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (4.68 + 14.4i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.88 + 9.46i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.381 + 1.17i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.29 + 3.99i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (9.13 + 6.63i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.65 - 1.51i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.12086048844011169591549376261, −11.01805465817382895033664493418, −10.08751410918501915107943833357, −8.799905812881292886187283640028, −7.905183346208876674056194323633, −7.15893049223922361634114907426, −6.15111993842000928854847772667, −4.52393850154239619390672709404, −2.13796329738566241573632540878, −0.56403961727030029652728028781,
2.66974058068507493105741347627, 4.01902816512589696258415839901, 5.79059138026337785555740669394, 6.73881210058043802555521636178, 8.247540507813705984435559392782, 8.861323383847795653005807846361, 10.28303425707104155076896056808, 10.77911108111962338740613026829, 11.28457116089312571222055765283, 12.53922034906677560459590002134
