| L(s) = 1 | + (−1.68 − 0.392i)3-s + 3.60i·5-s + 0.570i·7-s + (2.69 + 1.32i)9-s − 1.81·11-s − 4.81·13-s + (1.41 − 6.07i)15-s + 6.70i·17-s − i·19-s + (0.224 − 0.962i)21-s + 4.05·23-s − 7.98·25-s + (−4.01 − 3.29i)27-s + 3.76i·29-s + 7.33i·31-s + ⋯ |
| L(s) = 1 | + (−0.973 − 0.226i)3-s + 1.61i·5-s + 0.215i·7-s + (0.897 + 0.441i)9-s − 0.547·11-s − 1.33·13-s + (0.365 − 1.56i)15-s + 1.62i·17-s − 0.229i·19-s + (0.0489 − 0.210i)21-s + 0.844·23-s − 1.59·25-s + (−0.773 − 0.633i)27-s + 0.698i·29-s + 1.31i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3278701540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3278701540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.68 + 0.392i)T \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 - 3.60iT - 5T^{2} \) |
| 7 | \( 1 - 0.570iT - 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 - 6.70iT - 17T^{2} \) |
| 23 | \( 1 - 4.05T + 23T^{2} \) |
| 29 | \( 1 - 3.76iT - 29T^{2} \) |
| 31 | \( 1 - 7.33iT - 31T^{2} \) |
| 37 | \( 1 + 3.33T + 37T^{2} \) |
| 41 | \( 1 + 8.85iT - 41T^{2} \) |
| 43 | \( 1 + 9.05iT - 43T^{2} \) |
| 47 | \( 1 + 0.0706T + 47T^{2} \) |
| 53 | \( 1 + 2.91iT - 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 0.707iT - 67T^{2} \) |
| 71 | \( 1 + 4.66T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 1.85iT - 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 6.26iT - 89T^{2} \) |
| 97 | \( 1 + 9.29T + 97T^{2} \) |
| 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
−10.15978898449819238305794976422, −8.987331547451854172696416364544, −7.86699374310211613558483547887, −6.97704989640717641402198740055, −6.83920207024938266221424053030, −5.71121116215917290317084168824, −5.10304923480626835910339577015, −3.87556156020234899566323619997, −2.81834357530216372057057410759, −1.86010576121065740820091561934,
0.15268745302855789899508635749, 1.12311205525748513923931352161, 2.63458802841328522738835625723, 4.21441574348749560198783767210, 4.89592386350462037041405780013, 5.22085304174593137714589809664, 6.21905562003221656421913009770, 7.35951070664021523195791793543, 7.83884305066768192427008789091, 9.061959749940197254897557560762
