L(s) = 1 | + (1.58 + 2.73i)3-s + (1.41 − 2.44i)5-s + (−3.5 + 6.06i)9-s + (2.23 + 3.87i)11-s + 8.94·15-s + (2.12 + 3.67i)17-s + (1.58 − 2.73i)19-s + (4.47 − 7.74i)23-s + (−1.49 − 2.59i)25-s − 12.6·27-s − 6·29-s + (3.16 + 5.47i)31-s + (−7.07 + 12.2i)33-s + (−1 + 1.73i)37-s − 1.41·41-s + ⋯ |
L(s) = 1 | + (0.912 + 1.58i)3-s + (0.632 − 1.09i)5-s + (−1.16 + 2.02i)9-s + (0.674 + 1.16i)11-s + 2.30·15-s + (0.514 + 0.891i)17-s + (0.362 − 0.628i)19-s + (0.932 − 1.61i)23-s + (−0.299 − 0.519i)25-s − 2.43·27-s − 1.11·29-s + (0.567 + 0.983i)31-s + (−1.23 + 2.13i)33-s + (−0.164 + 0.284i)37-s − 0.220·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.699097183\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.699097183\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.58 - 2.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.41 + 2.44i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.23 - 3.87i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-2.12 - 3.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.58 + 2.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.47 + 7.74i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-3.16 - 5.47i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 + (3.16 - 5.47i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.58 - 2.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.24 - 7.34i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + (3.53 + 6.12i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.47 + 7.74i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 + (-4.94 + 8.57i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−9.427655897091823147169716506782, −9.020377453292069126542367055111, −8.480353282894065913476472719764, −7.44249207412411205815730657366, −6.19303470209642883679053749766, −4.97861153121658383644148293890, −4.73840207015896642258964876440, −3.82422413743045640042039148498, −2.75010099383416978428026667499, −1.56960609968003954505803017565,
1.03234074917102266609530775315, 2.06621495128793449009806790693, 3.07937336556493912510871914783, 3.52214764006437785361948887409, 5.60711152767621937364057070860, 6.14791302842374631208570614017, 7.01264113245769054428726041013, 7.49008799434143227405805251579, 8.271614723506847100450856034870, 9.252311457696686297598566275292