L(s) = 1 | + (1.12 − 0.648i)2-s + (0.752 + 2.80i)3-s + (−0.159 + 0.276i)4-s + (−1.50 + 0.867i)5-s + (2.66 + 2.66i)6-s + (−0.701 − 2.55i)7-s + 3.00i·8-s + (−4.72 + 2.72i)9-s + (−1.12 + 1.94i)10-s + (0.603 + 2.25i)11-s + (−0.895 − 0.239i)12-s + (3.55 − 3.55i)13-s + (−2.44 − 2.40i)14-s + (−3.56 − 3.56i)15-s + (1.63 + 2.82i)16-s + (1.30 − 0.348i)17-s + ⋯ |
L(s) = 1 | + (0.794 − 0.458i)2-s + (0.434 + 1.62i)3-s + (−0.0796 + 0.138i)4-s + (−0.672 + 0.388i)5-s + (1.08 + 1.08i)6-s + (−0.265 − 0.964i)7-s + 1.06i·8-s + (−1.57 + 0.909i)9-s + (−0.355 + 0.616i)10-s + (0.181 + 0.678i)11-s + (−0.258 − 0.0692i)12-s + (0.986 − 0.986i)13-s + (−0.652 − 0.644i)14-s + (−0.921 − 0.921i)15-s + (0.407 + 0.706i)16-s + (0.315 − 0.0846i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0376 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0376 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29308 + 1.24528i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29308 + 1.24528i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.701 + 2.55i)T \) |
| 41 | \( 1 + (-5.82 - 2.66i)T \) |
good | 2 | \( 1 + (-1.12 + 0.648i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.752 - 2.80i)T + (-2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (1.50 - 0.867i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.603 - 2.25i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-3.55 + 3.55i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.30 + 0.348i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.586 + 2.18i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.47 - 6.02i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.651 + 0.651i)T - 29iT^{2} \) |
| 31 | \( 1 + (-2.55 + 4.42i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.37 - 5.84i)T + (-18.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + 11.9iT - 43T^{2} \) |
| 47 | \( 1 + (-2.86 + 10.6i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.38 + 8.89i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.182 - 0.316i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.12 - 0.649i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (15.2 - 4.07i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (2.32 - 2.32i)T - 71iT^{2} \) |
| 73 | \( 1 + (4.26 + 2.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.101 - 0.0273i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 - 6.06T + 83T^{2} \) |
| 89 | \( 1 + (-2.82 - 0.756i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (0.441 + 0.441i)T + 97iT^{2} \) |
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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
−11.80625654384061965009413017101, −11.08100556441057019133386532440, −10.36059464831582951473639417156, −9.434116312184302118741464307407, −8.315335360926940018241614420155, −7.36250844055927643317458551825, −5.49761816545111661127423510242, −4.43016371766714331387389417477, −3.68051256823532809428381131744, −3.09997131734343058672034565635,
1.16319071117190837611555788981, 2.97755826727822412509614654453, 4.39684325381887555450311186104, 6.04847502826320633753389584701, 6.31613093935045565418893537816, 7.57502720770578935078889089866, 8.563297667965728724543098705847, 9.201347972651197969585365839323, 11.05109132379448579041673192069, 12.22349111598761513368468027273