| L(s) = 1 | + (−1.62 + 0.345i)2-s + (0.700 − 0.311i)4-s + (0.380 + 2.20i)5-s + (0.694 + 1.20i)7-s + (1.65 − 1.20i)8-s + (−1.38 − 3.45i)10-s + (4.44 − 0.945i)11-s + (−0.664 − 0.141i)13-s + (−1.54 − 1.71i)14-s + (−3.30 + 3.67i)16-s + (1.73 − 1.26i)17-s + (5.56 − 4.04i)19-s + (0.953 + 1.42i)20-s + (−6.90 + 3.07i)22-s + (0.0716 + 0.0796i)23-s + ⋯ |
| L(s) = 1 | + (−1.15 + 0.244i)2-s + (0.350 − 0.155i)4-s + (0.170 + 0.985i)5-s + (0.262 + 0.454i)7-s + (0.586 − 0.426i)8-s + (−0.436 − 1.09i)10-s + (1.34 − 0.285i)11-s + (−0.184 − 0.0391i)13-s + (−0.412 − 0.458i)14-s + (−0.827 + 0.918i)16-s + (0.420 − 0.305i)17-s + (1.27 − 0.928i)19-s + (0.213 + 0.318i)20-s + (−1.47 + 0.655i)22-s + (0.0149 + 0.0166i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.762911 + 0.506097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.762911 + 0.506097i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.380 - 2.20i)T \) |
| good | 2 | \( 1 + (1.62 - 0.345i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-0.694 - 1.20i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.44 + 0.945i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.664 + 0.141i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-1.73 + 1.26i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-5.56 + 4.04i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.0716 - 0.0796i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.214 + 2.04i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.675 - 6.43i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (0.886 + 2.72i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-5.31 - 1.12i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (0.613 + 1.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.23 - 11.7i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-1.07 - 0.781i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.69 + 0.360i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.69 + 0.784i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.48 - 14.1i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-10.9 - 7.96i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.83 - 5.63i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.938 + 8.92i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (16.5 + 7.37i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (2.33 - 7.18i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.13 - 10.8i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−10.46484640024769276599300003822, −9.534473862215475691029108633232, −9.138758534400465379542827540315, −8.102152084652542808277312092824, −7.20098389444385471557077797422, −6.62529820079047425046787729367, −5.43642587164986222744899563709, −4.02367134410087877194185114235, −2.78759643830104507764331296515, −1.21807737464589377787404870655,
0.939318557900853975578709107651, 1.79393725817672982330564795355, 3.80465854703123432241807355883, 4.78013551250141846597938208040, 5.83001592585832196066694798330, 7.18403839630068787537770202377, 7.981186826696697329043155332778, 8.719768939581477810659150025580, 9.647724501032545748289154130869, 9.863030217545460139802743234230
