| L(s) = 1 | − 26i·3-s + 22i·7-s − 433·9-s + 768·11-s + 46i·13-s + 378i·17-s + 1.10e3·19-s + 572·21-s − 1.98e3i·23-s + 4.94e3i·27-s + 5.61e3·29-s + 3.98e3·31-s − 1.99e4i·33-s − 142i·37-s + 1.19e3·39-s + ⋯ |
| L(s) = 1 | − 1.66i·3-s + 0.169i·7-s − 1.78·9-s + 1.91·11-s + 0.0754i·13-s + 0.317i·17-s + 0.699·19-s + 0.283·21-s − 0.782i·23-s + 1.30i·27-s + 1.23·29-s + 0.745·31-s − 3.19i·33-s − 0.0170i·37-s + 0.125·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.412377147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.412377147\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 26iT - 243T^{2} \) |
| 7 | \( 1 - 22iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 768T + 1.61e5T^{2} \) |
| 13 | \( 1 - 46iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 378iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.98e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.61e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 142iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.54e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.02e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.47e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.41e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.52e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.47e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.21e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.98e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.44e5iT - 8.58e9T^{2} \) |
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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
−10.15893822531685608235511401134, −8.908934309594006724504838483060, −8.323952509192753902379838900065, −7.08865899247764819822239076232, −6.62868528995988956864764096511, −5.71887296143189412864602388035, −4.12686890211326358050869934821, −2.70757420039120224875917508315, −1.52034074498381282858868796696, −0.73900969042252917842222002760,
1.09470075451311396386672800053, 3.02358819192208971380518445172, 3.95211888331383763600277715954, 4.67479299495870732628045380727, 5.80388654690700414337193586015, 6.88104052576497074417768955433, 8.299169096156638168414030160296, 9.325860564398023733496074891392, 9.643235704591954400963775149991, 10.65335929466759936337545312410
