| L(s) = 1 | + (1.24 + 1.20i)3-s − 5-s + 4.23i·7-s + (0.103 + 2.99i)9-s + 5.66·11-s + 6.93i·13-s + (−1.24 − 1.20i)15-s + 4.78i·17-s + 1.51·19-s + (−5.09 + 5.26i)21-s − 3.29i·23-s + 25-s + (−3.47 + 3.85i)27-s − 10.0i·29-s + 6.69i·31-s + ⋯ |
| L(s) = 1 | + (0.719 + 0.694i)3-s − 0.447·5-s + 1.59i·7-s + (0.0345 + 0.999i)9-s + 1.70·11-s + 1.92i·13-s + (−0.321 − 0.310i)15-s + 1.15i·17-s + 0.348·19-s + (−1.11 + 1.14i)21-s − 0.686i·23-s + 0.200·25-s + (−0.669 + 0.742i)27-s − 1.86i·29-s + 1.20i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.570596772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.570596772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.24 - 1.20i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (8.12 - 1.00i)T \) |
| good | 7 | \( 1 - 4.23iT - 7T^{2} \) |
| 11 | \( 1 - 5.66T + 11T^{2} \) |
| 13 | \( 1 - 6.93iT - 13T^{2} \) |
| 17 | \( 1 - 4.78iT - 17T^{2} \) |
| 19 | \( 1 - 1.51T + 19T^{2} \) |
| 23 | \( 1 + 3.29iT - 23T^{2} \) |
| 29 | \( 1 + 10.0iT - 29T^{2} \) |
| 31 | \( 1 - 6.69iT - 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 + 5.51iT - 43T^{2} \) |
| 47 | \( 1 - 7.87iT - 47T^{2} \) |
| 53 | \( 1 - 7.35T + 53T^{2} \) |
| 59 | \( 1 - 0.429iT - 59T^{2} \) |
| 61 | \( 1 + 8.83iT - 61T^{2} \) |
| 71 | \( 1 + 10.0iT - 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 4.49iT - 79T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 17.5iT - 97T^{2} \) |
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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
−8.691675781810097128601138469620, −8.440710332644442856296774614192, −7.33549771661572083499293094346, −6.38793151603367551935762098527, −5.93167601154328104337048972531, −4.64192869623645821110729217516, −4.20871045042363556604992200034, −3.44115383178361294157227258532, −2.33561580312347204892297562520, −1.67836077660737100264600616701,
0.78698690406316037220214851643, 1.15504782091659095213771250555, 2.74320833518677147950288893644, 3.58996494852731205859939216566, 3.96277675087705517692925603799, 5.10976127404253355601538996472, 6.16011173750117979725938415917, 7.05442333246212345584390027508, 7.37153297635346583814826671530, 7.940086634962093888487909518403
