| L(s) = 1 | − 0.874i·3-s + 1.23·5-s + 0.874i·7-s + 2.23·9-s − 3.70i·11-s + 2.82i·13-s − 1.08i·15-s − 6.53i·19-s + 0.763·21-s − 2.47·23-s − 3.47·25-s − 4.57i·27-s − 8.48i·29-s + 2.47·31-s − 3.23·33-s + ⋯ |
| L(s) = 1 | − 0.504i·3-s + 0.552·5-s + 0.330i·7-s + 0.745·9-s − 1.11i·11-s + 0.784i·13-s − 0.278i·15-s − 1.49i·19-s + 0.166·21-s − 0.515·23-s − 0.694·25-s − 0.880i·27-s − 1.57i·29-s + 0.444·31-s − 0.563·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.984802364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984802364\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + (-1 - 6.32i)T \) |
| good | 3 | \( 1 + 0.874iT - 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 0.874iT - 7T^{2} \) |
| 11 | \( 1 + 3.70iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6.53iT - 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 0.206iT - 47T^{2} \) |
| 53 | \( 1 + 0.667iT - 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 1.95iT - 67T^{2} \) |
| 71 | \( 1 + 8.94iT - 71T^{2} \) |
| 73 | \( 1 - 5.23T + 73T^{2} \) |
| 79 | \( 1 + 11.1iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 9.15iT - 97T^{2} \) |
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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
−8.714818171668583706382429516069, −7.924009109493232121426330437133, −7.12220323436696700669567497462, −6.31656716673730043364458581929, −5.85510199063789861832658179571, −4.73959586334252322438740871751, −3.94072566822198935331431624189, −2.68059380382042019424930072295, −1.92180549864752802914971929953, −0.68004793669570363202113693631,
1.32961954708234612427514251323, 2.24857935352656056718483144558, 3.60721465010598898242917529929, 4.18230128750803367412137230211, 5.17983020792122065264072264802, 5.78536036772468075981681145560, 6.86930263417226844191959325493, 7.45623762059115557212167182696, 8.256957760625411904503453489921, 9.248553034423684183745704843999
