L(s) = 1 | + (−1.99 − 0.177i)2-s − 1.73·3-s + (3.93 + 0.707i)4-s + (3.45 + 0.307i)6-s − 1.19·7-s + (−7.71 − 2.10i)8-s + 2.99·9-s + 8.22i·11-s + (−6.81 − 1.22i)12-s − 11.1i·13-s + (2.38 + 0.212i)14-s + (14.9 + 5.57i)16-s + 20.9i·17-s + (−5.97 − 0.533i)18-s − 27.9i·19-s + ⋯ |
L(s) = 1 | + (−0.996 − 0.0888i)2-s − 0.577·3-s + (0.984 + 0.176i)4-s + (0.575 + 0.0512i)6-s − 0.170·7-s + (−0.964 − 0.263i)8-s + 0.333·9-s + 0.747i·11-s + (−0.568 − 0.102i)12-s − 0.860i·13-s + (0.170 + 0.0151i)14-s + (0.937 + 0.348i)16-s + 1.23i·17-s + (−0.332 − 0.0296i)18-s − 1.47i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0757339 + 0.227938i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0757339 + 0.227938i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.99 + 0.177i)T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.19T + 49T^{2} \) |
| 11 | \( 1 - 8.22iT - 121T^{2} \) |
| 13 | \( 1 + 11.1iT - 169T^{2} \) |
| 17 | \( 1 - 20.9iT - 289T^{2} \) |
| 19 | \( 1 + 27.9iT - 361T^{2} \) |
| 23 | \( 1 + 9.48T + 529T^{2} \) |
| 29 | \( 1 + 40.4T + 841T^{2} \) |
| 31 | \( 1 - 55.3iT - 961T^{2} \) |
| 37 | \( 1 - 50.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 73.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 19.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 18.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 57.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 60.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 21.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 9.68T + 4.48e3T^{2} \) |
| 71 | \( 1 + 68.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 84.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 23.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 93.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 62.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 91.3iT - 9.40e3T^{2} \) |
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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
−11.74266500092055138766043515794, −10.70279621825824117526565271951, −10.18430115770000006481661134161, −9.131960211202274164125969237588, −8.128634463882730157677077235691, −7.09398127023627797624982647169, −6.25916370934023715373762864111, −4.97849872556877485422919400800, −3.25832046790607480518474828751, −1.61360058510688238617256953206,
0.16912129534301495380833936648, 1.90804292484850650856039147502, 3.65735030010786255916788920736, 5.43805220000366661080409369093, 6.32236603356265550847974089655, 7.32384782622717961463816430454, 8.286390050728492802468335428318, 9.440852797671677744961975507538, 10.02110158782863351918060550706, 11.32804147342642127286092791795