L(s) = 1 | − 6.34·3-s + 6.94i·5-s + 13.3·9-s + 2.01i·11-s − 11.3i·13-s − 44.0i·15-s + 71.5i·17-s + 81.8·19-s − 192. i·23-s + 76.7·25-s + 86.9·27-s − 92.2·29-s − 252.·31-s − 12.7i·33-s + 277.·37-s + ⋯ |
L(s) = 1 | − 1.22·3-s + 0.620i·5-s + 0.492·9-s + 0.0551i·11-s − 0.242i·13-s − 0.758i·15-s + 1.02i·17-s + 0.988·19-s − 1.74i·23-s + 0.614·25-s + 0.619·27-s − 0.590·29-s − 1.46·31-s − 0.0673i·33-s + 1.23·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8520684054\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8520684054\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 6.34T + 27T^{2} \) |
| 5 | \( 1 - 6.94iT - 125T^{2} \) |
| 11 | \( 1 - 2.01iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 11.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 71.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 81.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 192. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 92.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 277.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 276. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 418. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 416.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 184.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 148. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 180. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 456. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 874. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 428. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 337.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 35.0iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3iT - 9.12e5T^{2} \) |
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
−10.57664185332401646923475675577, −9.443105788056071074167331371017, −8.381884268269670361795935327286, −7.33285196950574081832424315324, −6.50550382387358380905847408619, −5.80436933643085990769454520965, −4.94558838249981157381511219472, −3.78530249510133930228166582496, −2.51479501340883220471885827317, −0.928804277201356343153907793326,
0.36217215043425742557695822885, 1.44783738433954088060115158626, 3.15047351761297301864910652534, 4.48694272511961303165969095111, 5.35302234529895197483006614817, 5.84032305375380235060956642258, 7.06562312262475115768783959881, 7.73371182668291502153100889225, 9.141619492538627517202371648255, 9.510576705923569915041102311812