L(s) = 1 | + (45 + 33.1i)5-s − 59.6i·7-s + 252·11-s + 119. i·13-s − 689. i·17-s + 220·19-s − 2.43e3i·23-s + (924. + 2.98e3i)25-s + 6.93e3·29-s − 6.75e3·31-s + (1.98e3 − 2.68e3i)35-s − 1.39e4i·37-s + 198·41-s − 417. i·43-s + 1.05e4i·47-s + ⋯ |
L(s) = 1 | + (0.804 + 0.593i)5-s − 0.460i·7-s + 0.627·11-s + 0.195i·13-s − 0.578i·17-s + 0.139·19-s − 0.959i·23-s + (0.295 + 0.955i)25-s + 1.53·29-s − 1.26·31-s + (0.273 − 0.370i)35-s − 1.67i·37-s + 0.0183·41-s − 0.0344i·43-s + 0.695i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.669182204\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.669182204\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-45 - 33.1i)T \) |
good | 7 | \( 1 + 59.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 252T + 1.61e5T^{2} \) |
| 13 | \( 1 - 119. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 689. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 220T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.43e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.39e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 198T + 1.15e8T^{2} \) |
| 43 | \( 1 + 417. iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 5.82e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.36e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.09e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.18e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.99e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.01e5iT - 8.58e9T^{2} \) |
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
−9.509824879243594892779965412343, −8.946115008338592699635025416803, −7.67347178334648190997557781558, −6.84166133765982907924657219422, −6.17321540506611517345470524712, −5.09067116896755134092841099086, −4.00282453510819375127940587782, −2.88818867417723993043865758414, −1.84908026162384818924121049253, −0.62026114143845586746120146031,
1.00945538302512885674216601645, 1.89354746575714583441968555692, 3.11554473428117728236681126926, 4.35770946052570680965135261146, 5.36520414027103962772436912856, 6.06745389451677500228793340953, 7.00659471411899864195927974180, 8.256038195068743085480254198922, 8.879766078992221920969633669209, 9.700666523348687745043247813654