| L(s) = 1 | − 0.255i·3-s + 0.657·5-s + 5.46i·7-s + 8.93·9-s − 7.62i·11-s + 23.5·13-s − 0.167i·15-s + 1.93·17-s − 4.35i·19-s + 1.39·21-s − 15.5i·23-s − 24.5·25-s − 4.58i·27-s − 21.8·29-s + 23.0i·31-s + ⋯ |
| L(s) = 1 | − 0.0851i·3-s + 0.131·5-s + 0.781i·7-s + 0.992·9-s − 0.692i·11-s + 1.81·13-s − 0.0111i·15-s + 0.114·17-s − 0.229i·19-s + 0.0664·21-s − 0.675i·23-s − 0.982·25-s − 0.169i·27-s − 0.753·29-s + 0.742i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.394108157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.394108157\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 3 | \( 1 + 0.255iT - 9T^{2} \) |
| 5 | \( 1 - 0.657T + 25T^{2} \) |
| 7 | \( 1 - 5.46iT - 49T^{2} \) |
| 11 | \( 1 + 7.62iT - 121T^{2} \) |
| 13 | \( 1 - 23.5T + 169T^{2} \) |
| 17 | \( 1 - 1.93T + 289T^{2} \) |
| 23 | \( 1 + 15.5iT - 529T^{2} \) |
| 29 | \( 1 + 21.8T + 841T^{2} \) |
| 31 | \( 1 - 23.0iT - 961T^{2} \) |
| 37 | \( 1 - 34.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 51.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 90.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 88.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 19.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 83.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 80.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 65.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 145. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 78.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 46.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 24.0T + 9.40e3T^{2} \) |
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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
−9.427078080877420889730883471745, −8.714531547486133759573856641730, −8.071248149676183473222328664282, −6.97700919734019048681885766171, −6.11739876155010139882542835990, −5.51530192857208132773658722075, −4.23507154359669132562230130966, −3.41274719770331364845891931290, −2.11027971148209999692094762776, −0.970726136519469994112295397032,
1.00120402701475092849840728375, 1.98222059885245578285085008629, 3.77904927993326749122684860471, 4.00745095336819721241649319621, 5.31920380872093819282721814055, 6.26446843417678573011497814577, 7.12648718736214217316277565217, 7.80335534323293022308681876752, 8.730416324010794990233791619394, 9.814803690233240603695886262271
