L(s) = 1 | + (−2.90 − 0.737i)3-s + (1.57 + 1.57i)5-s − 3.64i·7-s + (7.91 + 4.29i)9-s + (−1.19 − 1.19i)11-s + (−14.6 − 14.6i)13-s + (−3.41 − 5.74i)15-s − 28.0i·17-s + (−12.5 − 12.5i)19-s + (−2.69 + 10.6i)21-s + 29.2·23-s − 20.0i·25-s + (−19.8 − 18.3i)27-s + (−19.3 + 19.3i)29-s − 11.6·31-s + ⋯ |
L(s) = 1 | + (−0.969 − 0.245i)3-s + (0.314 + 0.314i)5-s − 0.520i·7-s + (0.878 + 0.476i)9-s + (−0.108 − 0.108i)11-s + (−1.12 − 1.12i)13-s + (−0.227 − 0.382i)15-s − 1.65i·17-s + (−0.662 − 0.662i)19-s + (−0.128 + 0.504i)21-s + 1.27·23-s − 0.801i·25-s + (−0.734 − 0.678i)27-s + (−0.667 + 0.667i)29-s − 0.375·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.494142 - 0.656161i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.494142 - 0.656161i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.90 + 0.737i)T \) |
good | 5 | \( 1 + (-1.57 - 1.57i)T + 25iT^{2} \) |
| 7 | \( 1 + 3.64iT - 49T^{2} \) |
| 11 | \( 1 + (1.19 + 1.19i)T + 121iT^{2} \) |
| 13 | \( 1 + (14.6 + 14.6i)T + 169iT^{2} \) |
| 17 | \( 1 + 28.0iT - 289T^{2} \) |
| 19 | \( 1 + (12.5 + 12.5i)T + 361iT^{2} \) |
| 23 | \( 1 - 29.2T + 529T^{2} \) |
| 29 | \( 1 + (19.3 - 19.3i)T - 841iT^{2} \) |
| 31 | \( 1 + 11.6T + 961T^{2} \) |
| 37 | \( 1 + (-0.771 + 0.771i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 25.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-40.5 + 40.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 50.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-46.2 - 46.2i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (22.7 + 22.7i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-12.7 - 12.7i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (10.6 + 10.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 122.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 15.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 51.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-37.8 + 37.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 5.45T + 7.92e3T^{2} \) |
| 97 | \( 1 + 81.1T + 9.40e3T^{2} \) |
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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
−12.03040874871685954268510630018, −10.91285748750018535120623087511, −10.34801297383506968498162904885, −9.212062118998842364775689651307, −7.49089693022083141753353579448, −6.95253383108946801632185667547, −5.56327729405500351497588658166, −4.67586827216423457361549689884, −2.69291988703037363756764567971, −0.53285264370671593875849076703,
1.83654079841516906502466647365, 4.06399266266121843652178026243, 5.21362026340395800426310175921, 6.16151369715554013142272405669, 7.27570281751374630005469028393, 8.803127105129405351159121273029, 9.709176456884568631446005601323, 10.67011804565854627578870792035, 11.67288589771642655442882795830, 12.50590682783813221240027597791