| L(s) = 1 | + 1.41·2-s + (−2.98 − 0.318i)3-s + 2.00·4-s + (−4.21 − 0.450i)6-s − 2.64i·7-s + 2.82·8-s + (8.79 + 1.89i)9-s + 2.46i·11-s + (−5.96 − 0.636i)12-s − 5.95i·13-s − 3.74i·14-s + 4.00·16-s − 17.1·17-s + (12.4 + 2.68i)18-s + 10.3·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (−0.994 − 0.106i)3-s + 0.500·4-s + (−0.703 − 0.0750i)6-s − 0.377i·7-s + 0.353·8-s + (0.977 + 0.211i)9-s + 0.223i·11-s + (−0.497 − 0.0530i)12-s − 0.458i·13-s − 0.267i·14-s + 0.250·16-s − 1.00·17-s + (0.691 + 0.149i)18-s + 0.545·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.917236659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917236659\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (2.98 + 0.318i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 - 2.46iT - 121T^{2} \) |
| 13 | \( 1 + 5.95iT - 169T^{2} \) |
| 17 | \( 1 + 17.1T + 289T^{2} \) |
| 19 | \( 1 - 10.3T + 361T^{2} \) |
| 23 | \( 1 - 25.4T + 529T^{2} \) |
| 29 | \( 1 + 2.22iT - 841T^{2} \) |
| 31 | \( 1 - 8.60T + 961T^{2} \) |
| 37 | \( 1 + 24.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 10.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 63.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 53.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 64.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 25.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 72.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 76.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 43.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 56.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 10.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 161. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 167. iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.848718431745170566904956449414, −8.725545237320064630402342501042, −7.45655804614729015679269639616, −6.95433574092608870686836060638, −6.05306796301588880435368469634, −5.17825553043411521008779039084, −4.49624400434911753499745605094, −3.43124210354311864390804514820, −2.00579819278696465802407815713, −0.59372774440887263855406575798,
1.18347691124667734924244736728, 2.60539573337855742130037401302, 3.87882683900010342446739014479, 4.82127532363460026183946511301, 5.44108957419453612279170240034, 6.49835416093368805001239115762, 6.90036170731410440770682656812, 8.117311227266988680719060788323, 9.210352776966606935559421922312, 10.01441423388427504162656745809
