| L(s) = 1 | + 1.55i·3-s + (−2.03 − 0.928i)5-s − 1.46·7-s + 0.593·9-s + 2.19i·11-s + 3.35·13-s + (1.44 − 3.15i)15-s + 3.77i·17-s + (−4.27 − 0.851i)19-s − 2.26i·21-s − 7.11·23-s + (3.27 + 3.77i)25-s + 5.57i·27-s − 6.65i·29-s + 1.93·31-s + ⋯ |
| L(s) = 1 | + 0.895i·3-s + (−0.909 − 0.415i)5-s − 0.552·7-s + 0.197·9-s + 0.661i·11-s + 0.931·13-s + (0.372 − 0.814i)15-s + 0.916i·17-s + (−0.980 − 0.195i)19-s − 0.495i·21-s − 1.48·23-s + (0.654 + 0.755i)25-s + 1.07i·27-s − 1.23i·29-s + 0.346·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2452591733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2452591733\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.03 + 0.928i)T \) |
| 19 | \( 1 + (4.27 + 0.851i)T \) |
| good | 3 | \( 1 - 1.55iT - 3T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 - 2.19iT - 11T^{2} \) |
| 13 | \( 1 - 3.35T + 13T^{2} \) |
| 17 | \( 1 - 3.77iT - 17T^{2} \) |
| 23 | \( 1 + 7.11T + 23T^{2} \) |
| 29 | \( 1 + 6.65iT - 29T^{2} \) |
| 31 | \( 1 - 1.93T + 31T^{2} \) |
| 37 | \( 1 + 4.72T + 37T^{2} \) |
| 41 | \( 1 + 7.88iT - 41T^{2} \) |
| 43 | \( 1 + 3.49T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 3.80T + 61T^{2} \) |
| 67 | \( 1 - 0.493iT - 67T^{2} \) |
| 71 | \( 1 + 1.93T + 71T^{2} \) |
| 73 | \( 1 - 11.0iT - 73T^{2} \) |
| 79 | \( 1 + 0.949T + 79T^{2} \) |
| 83 | \( 1 + 3.79T + 83T^{2} \) |
| 89 | \( 1 + 18.3iT - 89T^{2} \) |
| 97 | \( 1 + 0.649T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.992552391518491677877368728249, −9.130096536971012179438223148672, −8.412575888824323934926224467853, −7.67782626586761185211387538334, −6.62796049958116452764684215393, −5.82381710470652956519836755373, −4.54769360058806248759522032172, −4.11905843129689985859797372964, −3.39026537221260906036344874907, −1.76918114113374262412491335229,
0.097227394299152329788986388307, 1.51926885433376429050225351360, 2.91504567135896033592685221395, 3.71089731929959371121165744643, 4.70736696662278090153934434717, 6.22166527275147031808655273357, 6.47015429094475707312353128026, 7.45057825012924406455602220291, 8.082202567142366567121833460573, 8.775137574342403652225519133396
