| L(s) = 1 | + 29.2i·3-s − 44.5i·7-s − 611.·9-s − 349.·11-s + 255. i·13-s − 1.50e3i·17-s − 2.43e3·19-s + 1.30e3·21-s + 1.43e3i·23-s − 1.07e4i·27-s − 2.87e3·29-s − 8.94e3·31-s − 1.02e4i·33-s + 1.45e4i·37-s − 7.45e3·39-s + ⋯ |
| L(s) = 1 | + 1.87i·3-s − 0.343i·7-s − 2.51·9-s − 0.870·11-s + 0.418i·13-s − 1.26i·17-s − 1.54·19-s + 0.643·21-s + 0.565i·23-s − 2.84i·27-s − 0.634·29-s − 1.67·31-s − 1.63i·33-s + 1.74i·37-s − 0.784·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8339913101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8339913101\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 29.2iT - 243T^{2} \) |
| 7 | \( 1 + 44.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 349.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 255. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.50e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.43e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.43e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.94e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.45e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.50e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.34e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.44e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.83e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.53e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.98e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.92e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 981.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.83e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 3.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.13e4iT - 8.58e9T^{2} \) |
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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
−9.714880572816733519444576380591, −8.950932529899594350467042584524, −8.152257427977534344878882635813, −6.96921433212081338194729350222, −5.69792242188285629430105211748, −4.97986239639643515652963233944, −4.20950809463450620457655041234, −3.35621724918361390605958825375, −2.30541374757734125350585263866, −0.25435980667397859355865276757,
0.61331162634915997586957487636, 1.98273540301333650808755081011, 2.38229386557624791218506177154, 3.81304071756519267492800065342, 5.50159023943037333972605658776, 5.98750895180595878049720461062, 6.95082750696609796426362257113, 7.71244811133378138260190385937, 8.376994722813587168704633617432, 9.067503296554500128156884360205
