L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (−0.210 + 0.364i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s − 3.22·11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)13-s + 0.421·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−1.28 − 2.23i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s − 0.408·6-s + (−0.0796 + 0.137i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s − 0.316·10-s − 0.971·11-s + (0.144 + 0.249i)12-s + (−0.138 + 0.240i)13-s + 0.112·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.312 − 0.541i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6452040785\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6452040785\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (2.07 + 5.71i)T \) |
good | 7 | \( 1 + (0.210 - 0.364i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.28 + 2.23i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.11 + 3.65i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 5.60T + 23T^{2} \) |
| 29 | \( 1 + 7.24T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 41 | \( 1 + (-5.12 + 8.87i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 0.842T + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 + (-1.32 - 2.28i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.03 - 1.78i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.900 + 1.55i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.93 + 3.34i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.58 + 6.21i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 8.44T + 73T^{2} \) |
| 79 | \( 1 + (7.37 - 12.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.01 + 12.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.42 + 5.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.278226247460945045268698283985, −8.805526623324744967841700779569, −7.71596733874737497328443068962, −7.24917390472151777028204711578, −5.92709532320300527795502099489, −5.05322786047300883353350854386, −3.89081144611123849016734245845, −2.68289265648492365211391702278, −1.88952125598600164146787691791, −0.28704669500756113110904083768,
1.90086797633417812248260223145, 3.19842357936745317104341345196, 4.24522968139683531004663369749, 5.38668946527403644271120749970, 6.00214181799484897718108805663, 7.11291145660324562225490788406, 7.934930856426963869314644929642, 8.468788282654308975462801576023, 9.695245988738691956403243110582, 10.00063173005227696191315094867