L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (2.23 − 0.0466i)5-s + (−0.809 − 0.587i)6-s − 3.52i·7-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (−2.11 + 0.735i)10-s + (1.62 + 4.99i)11-s + (0.951 + 0.309i)12-s + (−0.588 − 0.191i)13-s + (1.08 + 3.34i)14-s + (1.35 + 1.78i)15-s + (0.309 − 0.951i)16-s + (2.02 − 2.78i)17-s + ⋯ |
L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.339 + 0.467i)3-s + (0.404 − 0.293i)4-s + (0.999 − 0.0208i)5-s + (−0.330 − 0.239i)6-s − 1.33i·7-s + (−0.207 + 0.286i)8-s + (−0.103 + 0.317i)9-s + (−0.667 + 0.232i)10-s + (0.489 + 1.50i)11-s + (0.274 + 0.0892i)12-s + (−0.163 − 0.0530i)13-s + (0.290 + 0.895i)14-s + (0.349 + 0.459i)15-s + (0.0772 − 0.237i)16-s + (0.491 − 0.676i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03233 + 0.219321i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03233 + 0.219321i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (-2.23 + 0.0466i)T \) |
good | 7 | \( 1 + 3.52iT - 7T^{2} \) |
| 11 | \( 1 + (-1.62 - 4.99i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.588 + 0.191i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.02 + 2.78i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.83 - 1.33i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (8.51 - 2.76i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.16 - 1.57i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (7.90 + 5.74i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.952 + 0.309i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.94 + 5.98i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 1.51iT - 43T^{2} \) |
| 47 | \( 1 + (6.27 + 8.63i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.325 - 0.447i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.0861 - 0.265i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 3.50i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (7.00 - 9.63i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (3.84 - 2.79i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-9.35 + 3.03i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.27 + 3.83i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.56 - 2.15i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-4.13 - 12.7i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (2.56 + 3.53i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−13.32407911235552733837067276265, −11.97471282169776881383486187279, −10.57201006734215199565741583667, −9.840287826060296677404417558749, −9.378899776135703111063011070545, −7.73096454491435614126478270379, −6.97578179596859319730062209694, −5.45280759704214018295464553695, −3.97036263420644048569315890827, −1.87744328217536809060412156640,
1.82178062140016157885121275801, 3.15059353067334317615240501357, 5.69283223682268175764259058830, 6.37042455923037834918133204637, 8.061709540707087754747429764543, 8.871151613178540654894506148576, 9.621429411489082631709238082761, 10.90001563583465098456301844079, 11.98034526068738761045359750917, 12.79874255017006211582642327824