| L(s) = 1 | + 1.41·2-s + 1.73i·3-s + 2.00·4-s + 2.44i·6-s + (−1.57 + 6.81i)7-s + 2.82·8-s − 2.99·9-s − 8.15·11-s + 3.46i·12-s − 14.6i·13-s + (−2.23 + 9.64i)14-s + 4.00·16-s + 5.81i·17-s − 4.24·18-s + 33.7i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (−0.225 + 0.974i)7-s + 0.353·8-s − 0.333·9-s − 0.741·11-s + 0.288i·12-s − 1.12i·13-s + (−0.159 + 0.688i)14-s + 0.250·16-s + 0.342i·17-s − 0.235·18-s + 1.77i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.360568544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.360568544\) |
| \(L(2)\) |
|
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 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.57 - 6.81i)T \) |
| good | 11 | \( 1 + 8.15T + 121T^{2} \) |
| 13 | \( 1 + 14.6iT - 169T^{2} \) |
| 17 | \( 1 - 5.81iT - 289T^{2} \) |
| 19 | \( 1 - 33.7iT - 361T^{2} \) |
| 23 | \( 1 + 37.2T + 529T^{2} \) |
| 29 | \( 1 - 9.25T + 841T^{2} \) |
| 31 | \( 1 - 19.2iT - 961T^{2} \) |
| 37 | \( 1 + 63.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.25iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 42.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 23.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 71.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 42.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.99T + 4.48e3T^{2} \) |
| 71 | \( 1 + 38.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 124. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 56.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 90.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 16.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 82.7iT - 9.40e3T^{2} \) |
| 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
−10.38298499037657173372025042367, −9.406143591582153476019559387196, −8.242758299723193534484510557389, −7.84401651131063164776950110646, −6.31852582551491561499721332863, −5.69356447371168131457315201319, −5.06944054282406466348285572192, −3.82561105067582603853009075332, −3.05936378202348026389421719975, −1.93802715423338939867490227815,
0.29634362754764275998668253146, 1.85579593192500491104401919015, 2.91195914652740602728697228911, 4.09803891980171431983522324301, 4.85725194752407330075475102115, 5.99302538150008474094884922099, 6.88793713887072282348245377341, 7.33602087452560147530225100725, 8.323655476974638286462001455287, 9.395504453898209785634169808624
