| L(s) = 1 | + (1.15 − 1.29i)3-s + 5-s + 2.00i·7-s + (−0.351 − 2.97i)9-s − 4.96·11-s + 1.09i·13-s + (1.15 − 1.29i)15-s + 2.56i·17-s − 2.48·19-s + (2.59 + 2.31i)21-s + 2.68i·23-s + 25-s + (−4.26 − 2.97i)27-s + 0.905i·29-s + 0.150i·31-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.747i)3-s + 0.447·5-s + 0.758i·7-s + (−0.117 − 0.993i)9-s − 1.49·11-s + 0.303i·13-s + (0.297 − 0.334i)15-s + 0.622i·17-s − 0.570·19-s + (0.567 + 0.504i)21-s + 0.560i·23-s + 0.200·25-s + (−0.820 − 0.572i)27-s + 0.168i·29-s + 0.0270i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5437043107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5437043107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.15 + 1.29i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (1.50 - 8.04i)T \) |
| good | 7 | \( 1 - 2.00iT - 7T^{2} \) |
| 11 | \( 1 + 4.96T + 11T^{2} \) |
| 13 | \( 1 - 1.09iT - 13T^{2} \) |
| 17 | \( 1 - 2.56iT - 17T^{2} \) |
| 19 | \( 1 + 2.48T + 19T^{2} \) |
| 23 | \( 1 - 2.68iT - 23T^{2} \) |
| 29 | \( 1 - 0.905iT - 29T^{2} \) |
| 31 | \( 1 - 0.150iT - 31T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 + 1.68iT - 43T^{2} \) |
| 47 | \( 1 + 13.1iT - 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 - 7.45iT - 59T^{2} \) |
| 61 | \( 1 - 11.3iT - 61T^{2} \) |
| 71 | \( 1 + 0.805iT - 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 0.585iT - 79T^{2} \) |
| 83 | \( 1 + 8.06iT - 83T^{2} \) |
| 89 | \( 1 - 18.0iT - 89T^{2} \) |
| 97 | \( 1 - 6.66iT - 97T^{2} \) |
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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
−8.684279255911489670752439889489, −8.085258819926882644921832319139, −7.24370307832624757532803788692, −6.63283300491808962255654843433, −5.67059591609909750186921613474, −5.26447014539496142096413289985, −3.97792859870444465789386203502, −3.00759517665366240113718712790, −2.29148139089704209583444907660, −1.58100137127421158131072114527,
0.12349323122562642721801250859, 1.82390403034074878700256642574, 2.77200598574801269924915181865, 3.39506751769715081976074761944, 4.52420759006144112194608373995, 4.97544040653179770691883143972, 5.80619873428382279718209345528, 6.85257714397540182229101625064, 7.62747746817699869089064536843, 8.205617272446792989287762868416
