| L(s) = 1 | + (−0.978 − 0.207i)2-s + (−1.68 − 0.390i)3-s + (0.913 + 0.406i)4-s + (1.82 − 1.29i)5-s + (1.56 + 0.732i)6-s + (1.61 − 2.78i)7-s + (−0.809 − 0.587i)8-s + (2.69 + 1.31i)9-s + (−2.05 + 0.883i)10-s + (3.49 + 0.742i)11-s + (−1.38 − 1.04i)12-s + (−6.84 + 1.45i)13-s + (−2.15 + 2.39i)14-s + (−3.58 + 1.46i)15-s + (0.669 + 0.743i)16-s + (3.96 + 2.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (−0.974 − 0.225i)3-s + (0.456 + 0.203i)4-s + (0.816 − 0.577i)5-s + (0.640 + 0.299i)6-s + (0.608 − 1.05i)7-s + (−0.286 − 0.207i)8-s + (0.898 + 0.439i)9-s + (−0.649 + 0.279i)10-s + (1.05 + 0.223i)11-s + (−0.399 − 0.301i)12-s + (−1.89 + 0.403i)13-s + (−0.576 + 0.639i)14-s + (−0.925 + 0.378i)15-s + (0.167 + 0.185i)16-s + (0.960 + 0.698i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.679296 - 0.589464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679296 - 0.589464i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (1.68 + 0.390i)T \) |
| 5 | \( 1 + (-1.82 + 1.29i)T \) |
| good | 7 | \( 1 + (-1.61 + 2.78i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.49 - 0.742i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (6.84 - 1.45i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.96 - 2.87i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.471 + 0.342i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-5.46 + 6.06i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.317 + 3.02i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.442 + 4.20i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.37 + 4.21i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (7.11 - 1.51i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (2.62 - 4.54i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.320 + 3.05i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-0.352 + 0.256i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-9.37 + 1.99i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (5.98 + 1.27i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.121 + 1.15i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-7.46 + 5.42i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.0385 - 0.118i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.936 + 8.90i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (8.64 - 3.84i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-3.99 - 12.2i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.0571 + 0.543i)T + (-94.8 + 20.1i)T^{2} \) |
| 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
−10.72802929779101922929906077092, −10.01479300788742106778752648215, −9.399583401634621466077246197030, −8.069859663488100130731624404053, −7.13646686465201511953865333283, −6.41612855657956361900546688390, −5.10118837233033970380306293904, −4.29842338614805636639649199145, −2.02564787468649176879441335909, −0.879862585798727334533858789382,
1.53355591965962600535643758042, 3.00406528582819900312325664648, 5.14930208143615242399648473641, 5.49041149167255879112055492935, 6.75570706923220201397532015067, 7.37533862750711027087951720032, 8.862303560442149285682125371515, 9.675770751015698384691922135719, 10.18099038123956308567322638508, 11.35273663270382547059061006961
