L(s) = 1 | + (−0.609 − 1.62i)3-s + (−2.24 + 1.29i)5-s + (2.53 + 0.751i)7-s + (−2.25 + 1.97i)9-s + (1.63 − 2.83i)11-s + 0.912·13-s + (3.47 + 2.85i)15-s + (2.39 − 4.14i)17-s + (−2.66 − 4.61i)19-s + (−0.326 − 4.57i)21-s + (4.45 − 2.57i)23-s + (0.870 − 1.50i)25-s + (4.57 + 2.45i)27-s − 1.35·29-s + (−8.18 − 4.72i)31-s + ⋯ |
L(s) = 1 | + (−0.351 − 0.936i)3-s + (−1.00 + 0.580i)5-s + (0.958 + 0.284i)7-s + (−0.752 + 0.658i)9-s + (0.493 − 0.855i)11-s + 0.253·13-s + (0.897 + 0.737i)15-s + (0.580 − 1.00i)17-s + (−0.611 − 1.05i)19-s + (−0.0712 − 0.997i)21-s + (0.929 − 0.536i)23-s + (0.174 − 0.301i)25-s + (0.881 + 0.473i)27-s − 0.252·29-s + (−1.47 − 0.848i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0326 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0326 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.752555 - 0.777518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.752555 - 0.777518i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.609 + 1.62i)T \) |
| 7 | \( 1 + (-2.53 - 0.751i)T \) |
good | 5 | \( 1 + (2.24 - 1.29i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.63 + 2.83i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.912T + 13T^{2} \) |
| 17 | \( 1 + (-2.39 + 4.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.66 + 4.61i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.45 + 2.57i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 + (8.18 + 4.72i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.59 + 0.922i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.91T + 41T^{2} \) |
| 43 | \( 1 + 8.00iT - 43T^{2} \) |
| 47 | \( 1 + (3.29 + 5.70i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.841 - 1.45i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.50 - 0.867i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.72 + 8.18i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.8 - 6.27i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.603iT - 71T^{2} \) |
| 73 | \( 1 + (1.29 + 0.746i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.0625 - 0.108i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.246iT - 83T^{2} \) |
| 89 | \( 1 + (-1.80 - 3.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.10iT - 97T^{2} \) |
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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
−10.87231523012788540055264927647, −9.122053310162273666610007242327, −8.403960961462822766900353667460, −7.50645677319093223844617394439, −6.98192528150683557731122475957, −5.85162595295861353834283950652, −4.88872585067824496081686506041, −3.54729724064595925945607354559, −2.33180771338702198120720207730, −0.67247282458021800038314761325,
1.43325343767091402917635357613, 3.59548262223836058396588850223, 4.24534994441780410609708001424, 5.00875722414094223602281126240, 6.06942835052424110746160710329, 7.42914408875188279843336344280, 8.197118479536771655443836797463, 8.968818472516545879145029157482, 9.930761208824099546211289597556, 10.87033157246335916576584092060