| L(s) = 1 | + (−1.18 − 2.32i)3-s + (0.966 + 2.01i)5-s + (0.0730 + 0.0730i)7-s + (−2.24 + 3.09i)9-s + (−0.496 − 0.683i)11-s + (0.983 + 6.20i)13-s + (3.54 − 4.64i)15-s + (−4.79 − 2.44i)17-s + (2.38 + 7.33i)19-s + (0.0833 − 0.256i)21-s + (−0.254 + 1.60i)23-s + (−3.13 + 3.89i)25-s + (2.13 + 0.337i)27-s + (8.34 + 2.71i)29-s + (−6.65 + 2.16i)31-s + ⋯ |
| L(s) = 1 | + (−0.684 − 1.34i)3-s + (0.432 + 0.901i)5-s + (0.0276 + 0.0276i)7-s + (−0.749 + 1.03i)9-s + (−0.149 − 0.206i)11-s + (0.272 + 1.72i)13-s + (0.915 − 1.19i)15-s + (−1.16 − 0.592i)17-s + (0.546 + 1.68i)19-s + (0.0181 − 0.0560i)21-s + (−0.0530 + 0.334i)23-s + (−0.626 + 0.779i)25-s + (0.410 + 0.0650i)27-s + (1.55 + 0.503i)29-s + (−1.19 + 0.388i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.984587 + 0.362540i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.984587 + 0.362540i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.966 - 2.01i)T \) |
| good | 3 | \( 1 + (1.18 + 2.32i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-0.0730 - 0.0730i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.496 + 0.683i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.983 - 6.20i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (4.79 + 2.44i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.38 - 7.33i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.254 - 1.60i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-8.34 - 2.71i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.65 - 2.16i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.0262 - 0.00416i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-4.26 - 3.09i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-1.83 + 1.83i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.86 + 2.99i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.916 + 0.467i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-5.94 - 4.32i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (1.59 - 1.15i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.166 - 0.326i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-6.56 - 2.13i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.655 + 0.103i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (1.26 - 3.88i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (6.93 + 3.53i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-9.44 - 12.9i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.11 - 2.19i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.57693952592304824468621349121, −9.546873762947249604693582293983, −8.564043571431006130191411217481, −7.40191198326252025794901220496, −6.84875499470089624132718409556, −6.24338548308869620059547150436, −5.37347784328672652133896374880, −3.88586121848673359853710959335, −2.39377897873565473613468174033, −1.48653673881592078767576635141,
0.58925407223284218165361835874, 2.65208838672684217451558249305, 4.07974226767765831828832462708, 4.83259347928931073022225422763, 5.48580118556515358812789547450, 6.32742647691056521644935178797, 7.76016316953778837494315410722, 8.808733981199198670621244998773, 9.354153187844043386299485038822, 10.30834286959137889228365802493
