L(s) = 1 | + (1.69 − 0.378i)3-s + (1.97 + 1.14i)5-s + (0.907 + 1.57i)7-s + (2.71 − 1.27i)9-s + (−4.24 + 2.44i)11-s + (−4.00 − 2.31i)13-s + (3.77 + 1.18i)15-s + 1.92·17-s − 2.12i·19-s + (2.12 + 2.31i)21-s + (1.15 − 2.00i)23-s + (0.101 + 0.175i)25-s + (4.10 − 3.18i)27-s + (−3.16 + 1.82i)29-s + (2.65 − 4.60i)31-s + ⋯ |
L(s) = 1 | + (0.975 − 0.218i)3-s + (0.883 + 0.510i)5-s + (0.343 + 0.594i)7-s + (0.904 − 0.426i)9-s + (−1.27 + 0.738i)11-s + (−1.11 − 0.641i)13-s + (0.973 + 0.304i)15-s + 0.467·17-s − 0.488i·19-s + (0.464 + 0.505i)21-s + (0.241 − 0.418i)23-s + (0.0203 + 0.0351i)25-s + (0.789 − 0.613i)27-s + (−0.587 + 0.339i)29-s + (0.477 − 0.826i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.88680 + 0.176110i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88680 + 0.176110i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 + 0.378i)T \) |
good | 5 | \( 1 + (-1.97 - 1.14i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.907 - 1.57i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.24 - 2.44i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.00 + 2.31i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 + 2.12iT - 19T^{2} \) |
| 23 | \( 1 + (-1.15 + 2.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.16 - 1.82i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.65 + 4.60i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.98iT - 37T^{2} \) |
| 41 | \( 1 + (2.36 - 4.09i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.20 - 1.27i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.02 - 3.49i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.95iT - 53T^{2} \) |
| 59 | \( 1 + (-3.05 - 1.76i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.71 - 0.991i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.72 + 4.46i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + (-4.97 - 8.61i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.12 + 1.80i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2.49T + 89T^{2} \) |
| 97 | \( 1 + (-6.99 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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
−12.09443839814195120373103881923, −10.54857390176125583041284815839, −9.942405910636860315530267683728, −9.074196699445853263641643967081, −7.86869965905694020485707663157, −7.24045466324832729587535594798, −5.82297437917006413322522909613, −4.72877896361688186998570540697, −2.81909774186464718483102533297, −2.20326605469527484861524068075,
1.78231978689694629461626038609, 3.13131395909933429885049138784, 4.63167866456645342483203556436, 5.53763829919929620229992196647, 7.16712585684777951059957234264, 8.019601679103979000287062860955, 8.948844339850066198277097492189, 9.948142941650537580346457606113, 10.43631351552081198865903840486, 11.85207638533630653688229142640