| L(s) = 1 | − 11.5i·3-s − 27.0i·7-s + 110.·9-s − 226.·11-s − 511. i·13-s + 387. i·17-s − 1.33e3·19-s − 311.·21-s + 545. i·23-s − 4.07e3i·27-s + 4.63e3·29-s − 2.99e3·31-s + 2.60e3i·33-s − 1.26e3i·37-s − 5.89e3·39-s + ⋯ |
| L(s) = 1 | − 0.739i·3-s − 0.208i·7-s + 0.453·9-s − 0.563·11-s − 0.839i·13-s + 0.325i·17-s − 0.848·19-s − 0.154·21-s + 0.214i·23-s − 1.07i·27-s + 1.02·29-s − 0.559·31-s + 0.416i·33-s − 0.151i·37-s − 0.620·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4987968354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4987968354\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 11.5iT - 243T^{2} \) |
| 7 | \( 1 + 27.0iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 226.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 511. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 387. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 545. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 4.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.71e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.65e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.30e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.89e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.93e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 9.06e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.55e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.01e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.32e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 4.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.05e4iT - 8.58e9T^{2} \) |
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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
−10.23212595707444045298753984591, −8.840316667138567391739350901246, −7.952250848493451539708666337690, −7.18149782651507435171991932914, −6.24655831961947912347772023758, −5.13284249925628783845851612813, −3.89774485908660560905560629565, −2.54182571481102836277657297841, −1.36185935200599537084277574108, −0.11960983469108833787289390937,
1.64551414249457844587245147104, 2.99789916525418945741066963265, 4.27634963470377505033110473711, 4.94753973206135619990763114506, 6.24870478256814141059083090402, 7.22171930189438933714770280347, 8.389841703845001929585952277786, 9.256039761107683314055704876857, 10.10226810724738888089744337475, 10.80299015461860045512354853275
