| L(s) = 1 | + i·2-s + 4.47·3-s + 3·4-s + 4.47i·6-s − 7i·7-s + 7i·8-s + 11.0·9-s + 2·11-s + 13.4·12-s − 13.4·13-s + 7·14-s + 5·16-s − 26.8·17-s + 11.0i·18-s − 13.4i·19-s + ⋯ |
| L(s) = 1 | + 0.5i·2-s + 1.49·3-s + 0.750·4-s + 0.745i·6-s − i·7-s + 0.875i·8-s + 1.22·9-s + 0.181·11-s + 1.11·12-s − 1.03·13-s + 0.5·14-s + 0.312·16-s − 1.57·17-s + 0.611i·18-s − 0.706i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.55672 + 0.603561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.55672 + 0.603561i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
| good | 2 | \( 1 - iT - 4T^{2} \) |
| 3 | \( 1 - 4.47T + 9T^{2} \) |
| 11 | \( 1 - 2T + 121T^{2} \) |
| 13 | \( 1 + 13.4T + 169T^{2} \) |
| 17 | \( 1 + 26.8T + 289T^{2} \) |
| 19 | \( 1 + 13.4iT - 361T^{2} \) |
| 23 | \( 1 - 26iT - 529T^{2} \) |
| 29 | \( 1 - 22T + 841T^{2} \) |
| 31 | \( 1 - 53.6iT - 961T^{2} \) |
| 37 | \( 1 + 14iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 26.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 34iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 40.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 93.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 14iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 62T + 5.04e3T^{2} \) |
| 73 | \( 1 + 53.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 38T + 6.24e3T^{2} \) |
| 83 | \( 1 - 40.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 26.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 26.8T + 9.40e3T^{2} \) |
| 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
−12.85411854043658106592920285849, −11.48639277403776992204870027076, −10.45363377325303386865089827337, −9.315060605194204751357081247292, −8.319700406401719198824630685979, −7.32834370949643430570017000516, −6.74770484874808900913817841216, −4.78189359579872758890509204845, −3.29121019107990596362902596636, −2.05522890169869190746273426585,
2.18245707412661714132154404108, 2.72588040412026452917495219098, 4.26063865504008912041474479826, 6.20054814170145077959704121991, 7.40386059078576089930030869824, 8.480733101903409544576083552726, 9.332912338746479590423352072395, 10.27682438390313339155384600267, 11.55501605361846292150637099424, 12.41394009169006194378078509857
