L(s) = 1 | + i·2-s + (1.45 − 0.942i)3-s − 4-s + (−0.724 + 1.25i)5-s + (0.942 + 1.45i)6-s − i·8-s + (1.22 − 2.73i)9-s + (−1.25 − 0.724i)10-s + (1.21 − 0.698i)11-s + (−1.45 + 0.942i)12-s + (3.03 − 1.75i)13-s + (0.129 + 2.50i)15-s + 16-s + (3.95 − 6.84i)17-s + (2.73 + 1.22i)18-s + (−3.61 + 2.08i)19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.839 − 0.544i)3-s − 0.5·4-s + (−0.324 + 0.561i)5-s + (0.384 + 0.593i)6-s − 0.353i·8-s + (0.408 − 0.912i)9-s + (−0.396 − 0.229i)10-s + (0.364 − 0.210i)11-s + (−0.419 + 0.272i)12-s + (0.841 − 0.485i)13-s + (0.0334 + 0.647i)15-s + 0.250·16-s + (0.958 − 1.66i)17-s + (0.645 + 0.288i)18-s + (−0.828 + 0.478i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.99637 + 0.263717i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99637 + 0.263717i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-1.45 + 0.942i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.724 - 1.25i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 0.698i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.03 + 1.75i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.95 + 6.84i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.61 - 2.08i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.13 - 1.80i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.06 - 2.34i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 0.917iT - 31T^{2} \) |
| 37 | \( 1 + (-2.14 - 3.70i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.343 - 0.595i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.01 - 10.4i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8.31T + 47T^{2} \) |
| 53 | \( 1 + (5.16 + 2.98i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 9.44T + 59T^{2} \) |
| 61 | \( 1 + 9.85iT - 61T^{2} \) |
| 67 | \( 1 + 2.97T + 67T^{2} \) |
| 71 | \( 1 + 12.9iT - 71T^{2} \) |
| 73 | \( 1 + (9.79 + 5.65i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + (4.11 - 7.12i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.533 + 0.923i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.9 + 6.33i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.886069657121275618624027655571, −9.143461484824168156091651197383, −8.284473076098726555042359169633, −7.64470753116015217574581617846, −6.85972065800008290050157120053, −6.15001044500944277288465070635, −4.91574841531944548957438214559, −3.58143009438042716782768537999, −2.94012884041832907731370788205, −1.11543763667541119955679659662,
1.34884787164658893912004039400, 2.59035948113490285820147911557, 3.92627628332774425747705730825, 4.20776000284911057985239539314, 5.44885663806712581655204144773, 6.71776237928705291427862507358, 8.034641286093251996816606390665, 8.644084978450844676695511270250, 9.101420804533577646583676054812, 10.30875580931741481079806622612