L(s) = 1 | + (−0.707 + 1.22i)2-s + (2.73 − 1.58i)3-s + (−0.999 − 1.73i)4-s + (1.93 + 1.11i)5-s + 4.47i·6-s + 2.82·8-s + (3.5 − 6.06i)9-s + (−2.73 + 1.58i)10-s + (−5.47 − 3.16i)12-s + 7.07·15-s + (−2.00 + 3.46i)16-s + (4.94 + 8.57i)18-s − 4.47i·20-s + (0.707 − 1.22i)23-s + (7.74 − 4.47i)24-s + (2.5 + 4.33i)25-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)2-s + (1.58 − 0.912i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.499i)5-s + 1.82i·6-s + 0.999·8-s + (1.16 − 2.02i)9-s + (−0.866 + 0.499i)10-s + (−1.58 − 0.912i)12-s + 1.82·15-s + (−0.500 + 0.866i)16-s + (1.16 + 2.02i)18-s − 0.999i·20-s + (0.147 − 0.255i)23-s + (1.58 − 0.912i)24-s + (0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.40083 + 0.0761023i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.40083 + 0.0761023i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 5 | \( 1 + (-1.93 - 1.11i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-2.73 + 1.58i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 1.22i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + (8.21 + 4.74i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.6 - 6.70i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.12 + 3.67i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.48iT - 83T^{2} \) |
| 89 | \( 1 + (15.4 + 8.94i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.829461920886955045152997678667, −8.857025871989031973701001182830, −8.403450093910227816524126902787, −7.51555602433784085954370009797, −6.76637647105789483971711623913, −6.24512160777643899005716038148, −4.90796835516566840985453455375, −3.42879814922958629516428684232, −2.33429581462068515509876420701, −1.33506342659219661633388930645,
1.58949578365395893218380429992, 2.55844113035015102804332118387, 3.40083022737811055333015798002, 4.41481704322164131783136466560, 5.18330198356974332050970967621, 6.90022358653548268968425510193, 8.173042283986803624317543694768, 8.481538995078714179800696192728, 9.334046179676368082135534947080, 9.860954786236118283798075960602