| L(s) = 1 | + (−0.134 + 1.79i)2-s + (−1.31 + 0.896i)3-s + (−1.23 − 0.186i)4-s + (−0.368 + 1.61i)5-s + (−1.43 − 2.48i)6-s + (1.12 + 2.39i)7-s + (−0.300 + 1.31i)8-s + (−0.170 + 0.435i)9-s + (−2.85 − 0.879i)10-s + (1.50 − 3.83i)11-s + (1.79 − 0.863i)12-s + (0.378 + 0.351i)13-s + (−4.45 + 1.70i)14-s + (−0.962 − 2.45i)15-s + (−4.71 − 1.45i)16-s + (−1.14 − 5.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.0952 + 1.27i)2-s + (−0.759 + 0.517i)3-s + (−0.618 − 0.0931i)4-s + (−0.164 + 0.721i)5-s + (−0.585 − 1.01i)6-s + (0.426 + 0.904i)7-s + (−0.106 + 0.466i)8-s + (−0.0569 + 0.145i)9-s + (−0.902 − 0.278i)10-s + (0.453 − 1.15i)11-s + (0.517 − 0.249i)12-s + (0.105 + 0.0974i)13-s + (−1.19 + 0.455i)14-s + (−0.248 − 0.633i)15-s + (−1.17 − 0.363i)16-s + (−0.278 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.221723 - 0.846192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.221723 - 0.846192i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.12 - 2.39i)T \) |
| 43 | \( 1 + (-5.61 - 3.38i)T \) |
| good | 2 | \( 1 + (0.134 - 1.79i)T + (-1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (1.31 - 0.896i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.368 - 1.61i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-1.50 + 3.83i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.378 - 0.351i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (1.14 + 5.02i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (2.52 - 3.16i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (-0.160 - 0.201i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (0.176 - 2.35i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (0.0295 - 0.394i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 - 8.00T + 37T^{2} \) |
| 41 | \( 1 + (5.84 - 2.81i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-0.0352 - 0.0898i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-8.97 - 2.76i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (5.62 - 5.21i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.906 - 12.0i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-2.22 - 5.67i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (5.37 + 13.6i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (1.72 - 7.54i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + (3.37 + 5.85i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.299 + 3.99i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-6.07 - 2.92i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-2.22 - 2.79i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.83232828203590008159552650086, −11.37921530172781109162956627747, −10.60085581868342296116199013591, −9.148252326972065187011651444240, −8.369745971517663140853524067805, −7.32879040195245969921931687593, −6.20079044999210751571433939059, −5.66726112723358643158546688270, −4.60720486876152722048488481061, −2.76098798302535104441670204208,
0.74569816251269139840566626496, 1.91454373149421907752372299032, 3.87182248861602919284929592376, 4.69998104197474933952982869029, 6.33712909736964660622891034197, 7.18031582768115595825158791386, 8.551303321086629515187349183775, 9.600927199719187709323853281968, 10.58544378128151135744598833345, 11.25237363686661503911274159386
