| L(s) = 1 | − 1.61i·2-s − 0.618·4-s + (−2.14 + 0.618i)5-s − 3.32i·7-s − 2.23i·8-s + (1.00 + 3.47i)10-s − 1.14·11-s − 1.85i·13-s − 5.38·14-s − 4.85·16-s + 2.67i·17-s + 1.82·19-s + (1.32 − 0.381i)20-s + 1.85i·22-s + 0.420i·23-s + ⋯ |
| L(s) = 1 | − 1.14i·2-s − 0.309·4-s + (−0.961 + 0.276i)5-s − 1.25i·7-s − 0.790i·8-s + (0.316 + 1.09i)10-s − 0.346·11-s − 0.515i·13-s − 1.43·14-s − 1.21·16-s + 0.649i·17-s + 0.417·19-s + (0.296 − 0.0854i)20-s + 0.396i·22-s + 0.0876i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5202072353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5202072353\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.14 - 0.618i)T \) |
| 37 | \( 1 + iT \) |
| good | 2 | \( 1 + 1.61iT - 2T^{2} \) |
| 7 | \( 1 + 3.32iT - 7T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 + 1.85iT - 13T^{2} \) |
| 17 | \( 1 - 2.67iT - 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 - 0.420iT - 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 6.00T + 31T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 - 8.24iT - 43T^{2} \) |
| 47 | \( 1 + 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 1.64iT - 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 + 0.294T + 61T^{2} \) |
| 67 | \( 1 - 4.44iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 9.07iT - 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 + 0.115iT - 83T^{2} \) |
| 89 | \( 1 + 5.51T + 89T^{2} \) |
| 97 | \( 1 - 1.38iT - 97T^{2} \) |
| 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
−8.998216905180205230583378052290, −7.79675242492845198771863128868, −7.42891998573941330919570757653, −6.58545731073783935407777419783, −5.31093627274693561174179207661, −3.97314723869895762634641918528, −3.78672968216908512427443497482, −2.74277690554082805318836808837, −1.40997373720139852725353087427, −0.19588841274555955649169720184,
2.01022208080580972149778223828, 3.12931587197411921259633278597, 4.37985453070172487015361795866, 5.32550333854550162550669551755, 5.77795171270820322174389293910, 6.88557519664510331669947751565, 7.53176010086055411087117349349, 8.122578295236300518072062594077, 9.059743836777965714307245699282, 9.328017317314940791087834472593
