| L(s) = 1 | − 1.41·2-s + (−2.16 − 2.07i)3-s + 2.00·4-s + (3.06 + 2.93i)6-s + 2.64i·7-s − 2.82·8-s + (0.400 + 8.99i)9-s + 21.5i·11-s + (−4.33 − 4.14i)12-s + 15.4i·13-s − 3.74i·14-s + 4.00·16-s − 20.3·17-s + (−0.566 − 12.7i)18-s + 5.86·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.722 − 0.691i)3-s + 0.500·4-s + (0.511 + 0.488i)6-s + 0.377i·7-s − 0.353·8-s + (0.0445 + 0.999i)9-s + 1.96i·11-s + (−0.361 − 0.345i)12-s + 1.19i·13-s − 0.267i·14-s + 0.250·16-s − 1.19·17-s + (−0.0314 − 0.706i)18-s + 0.308·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06328446510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06328446510\) |
| \(L(2)\) |
|
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 | \( 1 + (2.16 + 2.07i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 - 21.5iT - 121T^{2} \) |
| 13 | \( 1 - 15.4iT - 169T^{2} \) |
| 17 | \( 1 + 20.3T + 289T^{2} \) |
| 19 | \( 1 - 5.86T + 361T^{2} \) |
| 23 | \( 1 + 11.3T + 529T^{2} \) |
| 29 | \( 1 - 3.43iT - 841T^{2} \) |
| 31 | \( 1 + 21.0T + 961T^{2} \) |
| 37 | \( 1 + 58.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 36.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 40.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 110. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 41.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 109. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 14.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 19.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 122.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 74.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 92.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 5.71iT - 9.40e3T^{2} \) |
| 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
−10.08137846069276897422260143246, −9.396612011444271937236440688587, −8.566441539166401429288620435879, −7.40655330646250306050980888895, −7.02459050792043491660964295878, −6.22972928966839765734791762859, −5.09272652197893074362280691030, −4.20754066730697457673556856735, −2.20693184413756301494755845719, −1.78408981820369419905781329301,
0.03222714469056710278438162916, 0.993391150772368918017674146227, 2.93541606865936906019085872802, 3.77718413570621329735095450276, 5.06164269752802564414588751379, 5.94963811111605108018924631679, 6.54830985047485470500202079691, 7.78456395592972928646923976792, 8.557485355412783535405273972826, 9.278990795273231600270624329232
