L(s) = 1 | + (−1.20 + 2.09i)2-s + (0.207 + 0.358i)3-s + (−1.91 − 3.31i)4-s + (−0.5 + 0.866i)5-s − 6-s + (2.62 − 0.358i)7-s + 4.41·8-s + (1.41 − 2.44i)9-s + (−1.20 − 2.09i)10-s + (0.414 + 0.717i)11-s + (0.792 − 1.37i)12-s − 4.82·13-s + (−2.41 + 5.91i)14-s − 0.414·15-s + (−1.49 + 2.59i)16-s + (−2.41 − 4.18i)17-s + ⋯ |
L(s) = 1 | + (−0.853 + 1.47i)2-s + (0.119 + 0.207i)3-s + (−0.957 − 1.65i)4-s + (−0.223 + 0.387i)5-s − 0.408·6-s + (0.990 − 0.135i)7-s + 1.56·8-s + (0.471 − 0.816i)9-s + (−0.381 − 0.661i)10-s + (0.124 + 0.216i)11-s + (0.228 − 0.396i)12-s − 1.33·13-s + (−0.645 + 1.58i)14-s − 0.106·15-s + (−0.374 + 0.649i)16-s + (−0.585 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.332372 + 0.406249i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.332372 + 0.406249i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.62 + 0.358i)T \) |
good | 2 | \( 1 + (1.20 - 2.09i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.207 - 0.358i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.414 - 0.717i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + (2.41 + 4.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.207 - 0.358i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.585 - 1.01i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.24 + 3.88i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.74 - 4.75i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.79 + 8.30i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 + (-0.414 - 0.717i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.41 - 12.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + (-4.32 + 7.49i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−17.02480970100776236278536054659, −15.69992099708420282197172482412, −14.90077895367364449460824843851, −14.16121949319904904757740790423, −11.99638652126369389252993355844, −10.22993831872043016925185227527, −9.094445240241736721926590931109, −7.71263000181428818990928551283, −6.74888324605567244040068521281, −4.84238648889317671558984205087,
2.08360306400790261273040738088, 4.55097300064152681458022245999, 7.74477049843809121485353588998, 8.763975966268183044027113514548, 10.20955880281173352222636554235, 11.26832080173664195164105518117, 12.31334337485115713826923228316, 13.38925708501227849862390675718, 15.05643379729391985440743834530, 16.89170553862582393734529839141