L(s) = 1 | − 1.47i·2-s − 3.55i·3-s + 1.82·4-s − 5.24·6-s − 7.05·7-s − 8.58i·8-s − 3.65·9-s + 2.24·11-s − 6.50i·12-s − 6.86i·13-s + 10.4i·14-s − 5.34·16-s − 11.1·17-s + 5.38i·18-s + (−15.8 − 10.4i)19-s + ⋯ |
L(s) = 1 | − 0.736i·2-s − 1.18i·3-s + 0.457·4-s − 0.873·6-s − 1.00·7-s − 1.07i·8-s − 0.406·9-s + 0.203·11-s − 0.542i·12-s − 0.527i·13-s + 0.743i·14-s − 0.333·16-s − 0.658·17-s + 0.299i·18-s + (−0.836 − 0.547i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.311854181\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311854181\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (15.8 + 10.4i)T \) |
good | 2 | \( 1 + 1.47iT - 4T^{2} \) |
| 3 | \( 1 + 3.55iT - 9T^{2} \) |
| 7 | \( 1 + 7.05T + 49T^{2} \) |
| 11 | \( 1 - 2.24T + 121T^{2} \) |
| 13 | \( 1 + 6.86iT - 169T^{2} \) |
| 17 | \( 1 + 11.1T + 289T^{2} \) |
| 23 | \( 1 + 21.1T + 529T^{2} \) |
| 29 | \( 1 - 45.9iT - 841T^{2} \) |
| 31 | \( 1 + 14.7iT - 961T^{2} \) |
| 37 | \( 1 - 25.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 64.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 34.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 20.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 69.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 23.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 79.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 26.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 109. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 124.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 85.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 35.5iT - 9.40e3T^{2} \) |
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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
−10.44945379707104132422363945427, −9.548247629813159464749365439680, −8.444041880743617132848605891630, −7.21260956355013353256722654407, −6.73355811247475592171508615744, −5.88365287717323976359747528520, −4.03887454516569796823110450999, −2.83688323490417857157973504263, −1.88412285438986453679976859910, −0.49283892928324950187905616457,
2.30690443968567608658763565727, 3.72277827033392470177323967147, 4.60700038087051305782348125533, 5.96068719067938272004881276403, 6.49082185011748052251028419121, 7.61802169486997077622078541602, 8.713684420407269065604431986256, 9.572964667004823729178424210461, 10.31272746582851090327148718295, 11.13838751462809542098245645076