L(s) = 1 | + 2.92·2-s − 2.02·3-s + 4.57·4-s − 5.92·6-s − 8.32i·7-s + 1.68·8-s − 4.90·9-s − 17.3·11-s − 9.25·12-s − 5.27·13-s − 24.3i·14-s − 13.3·16-s + 19.0i·17-s − 14.3·18-s + (11.4 − 15.1i)19-s + ⋯ |
L(s) = 1 | + 1.46·2-s − 0.674·3-s + 1.14·4-s − 0.987·6-s − 1.18i·7-s + 0.211·8-s − 0.545·9-s − 1.57·11-s − 0.771·12-s − 0.405·13-s − 1.74i·14-s − 0.834·16-s + 1.12i·17-s − 0.798·18-s + (0.601 − 0.798i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9522396143\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9522396143\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-11.4 + 15.1i)T \) |
good | 2 | \( 1 - 2.92T + 4T^{2} \) |
| 3 | \( 1 + 2.02T + 9T^{2} \) |
| 7 | \( 1 + 8.32iT - 49T^{2} \) |
| 11 | \( 1 + 17.3T + 121T^{2} \) |
| 13 | \( 1 + 5.27T + 169T^{2} \) |
| 17 | \( 1 - 19.0iT - 289T^{2} \) |
| 23 | \( 1 + 3.32iT - 529T^{2} \) |
| 29 | \( 1 + 37.4iT - 841T^{2} \) |
| 31 | \( 1 - 2.86iT - 961T^{2} \) |
| 37 | \( 1 + 33.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 20.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 33.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 37.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 32.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 6.58T + 3.72e3T^{2} \) |
| 67 | \( 1 - 112.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 110. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 134. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 48.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 44.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 173. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 50.6T + 9.40e3T^{2} \) |
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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
−10.78391988543806691735736511642, −9.933722746513353594972495597219, −8.369147579237597942820168330632, −7.33200512470299100638036352205, −6.37175439788343431122019869386, −5.42094795194750473308990227976, −4.78648853935656548281946795489, −3.68726874125634701377765352510, −2.53517112711080425485725528647, −0.23506160868411449556061648286,
2.47106220586556873036161719870, 3.20996108225112757316169932119, 5.00442685251702936809253819687, 5.27235320784406179000097946354, 6.00718962806656820859278081747, 7.18925422209640986136413831990, 8.398778706870864697315025952640, 9.470529762127697630742346975927, 10.69475333764912086182915248002, 11.52451611264985153146035792018