| L(s) = 1 | − 7.81i·2-s − 45.1·4-s − 21.3·5-s − 30.7·7-s + 227. i·8-s + 166. i·10-s + 207. i·11-s − 206. i·13-s + 240. i·14-s + 1.05e3·16-s + 391.·17-s − 321.·19-s + 962.·20-s + 1.62e3·22-s + 287. i·23-s + ⋯ |
| L(s) = 1 | − 1.95i·2-s − 2.82·4-s − 0.852·5-s − 0.627·7-s + 3.55i·8-s + 1.66i·10-s + 1.71i·11-s − 1.21i·13-s + 1.22i·14-s + 4.13·16-s + 1.35·17-s − 0.890·19-s + 2.40·20-s + 3.35·22-s + 0.544i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0942i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6782517987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6782517987\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-3.46e3 + 327. i)T \) |
| good | 2 | \( 1 + 7.81iT - 16T^{2} \) |
| 5 | \( 1 + 21.3T + 625T^{2} \) |
| 7 | \( 1 + 30.7T + 2.40e3T^{2} \) |
| 11 | \( 1 - 207. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 206. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 391.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 321.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 287. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.02e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 560. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.25e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.52e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.15e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 245. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.14e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 518. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.49e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 9.31e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 8.00e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 5.13e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.34e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.14e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 5.32e3iT - 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.860367187288364106871658369758, −9.549871201355333886328011038293, −8.180283955160574290273858907292, −7.54037167433932495254254830368, −5.65262538448136654496401299866, −4.55163430123180778568657249741, −3.72762183577280430953796228509, −2.87534168011470932467604003525, −1.65632428728655744587224877212, −0.33176932404686046710915504213,
0.61485511977552516471593476889, 3.53662889196341875108127013686, 4.12883028216502057421509066197, 5.49641381240288242373004128703, 6.15990818956370466209055047837, 7.04963612275318627138898105427, 7.84899667508357295184785427151, 8.663138455192379563532404534841, 9.223905087881045135115818521155, 10.39125590947268415653339235871
