| L(s) = 1 | + (−1.78 − 1.35i)5-s − 6.16i·11-s − 2.70i·13-s − 4.12·17-s − 8.68·19-s + 9.56·23-s + (1.34 + 4.81i)25-s − 6.92i·29-s + 9.63i·41-s + 13.0i·43-s − 7·49-s + (−8.34 + 10.9i)55-s + (−3.65 + 4.81i)65-s − 14.2i·67-s + 3.46i·71-s + ⋯ |
| L(s) = 1 | + (−0.796 − 0.604i)5-s − 1.85i·11-s − 0.750i·13-s − 0.999·17-s − 1.99·19-s + 1.99·23-s + (0.268 + 0.963i)25-s − 1.28i·29-s + 1.50i·41-s + 1.99i·43-s − 49-s + (−1.12 + 1.48i)55-s + (−0.453 + 0.597i)65-s − 1.74i·67-s + 0.411i·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2659343595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2659343595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.78 + 1.35i)T \) |
| 17 | \( 1 + 4.12T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 6.16iT - 11T^{2} \) |
| 13 | \( 1 + 2.70iT - 13T^{2} \) |
| 19 | \( 1 + 8.68T + 19T^{2} \) |
| 23 | \( 1 - 9.56T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 9.63iT - 41T^{2} \) |
| 43 | \( 1 - 13.0iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14.2iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
| 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
−8.284308368980384645026364352340, −7.81986341615186697237558629929, −6.55494323481716201750615663597, −6.13734915091694808951637060609, −5.01516292986077162401590287643, −4.40706291142784157534631711919, −3.43134121944157420616735387518, −2.67699638829507275951828753522, −1.08661312637316186209183116001, −0.091575499840399138545619508306,
1.82974375113398991440747564875, 2.56817093577911502900307292309, 3.82326956288301060026763753843, 4.43422495860349349561505761004, 5.09166436976419826983338647847, 6.53207036752919600127333847924, 7.01038534653349577297279247710, 7.31741201632856169168869704590, 8.690401053976915459325390853929, 8.853746953838048189384789878039
