L(s) = 1 | + (−0.730 − 1.26i)2-s + (−0.0665 + 0.115i)4-s + (−0.296 − 0.514i)5-s + (2.32 − 1.26i)7-s − 2.72·8-s + (−0.433 + 0.750i)10-s + (2.23 − 3.86i)11-s − 4.51·13-s + (−3.29 − 2.01i)14-s + (2.12 + 3.67i)16-s + (−0.136 + 0.236i)17-s + (−1.43 − 2.48i)19-s + 0.0789·20-s − 6.51·22-s + (2.52 + 4.37i)23-s + ⋯ |
L(s) = 1 | + (−0.516 − 0.894i)2-s + (−0.0332 + 0.0576i)4-s + (−0.132 − 0.229i)5-s + (0.878 − 0.478i)7-s − 0.964·8-s + (−0.137 + 0.237i)10-s + (0.672 − 1.16i)11-s − 1.25·13-s + (−0.881 − 0.538i)14-s + (0.531 + 0.919i)16-s + (−0.0331 + 0.0574i)17-s + (−0.328 − 0.569i)19-s + 0.0176·20-s − 1.38·22-s + (0.526 + 0.912i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.424858 - 0.817403i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.424858 - 0.817403i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.32 + 1.26i)T \) |
good | 2 | \( 1 + (0.730 + 1.26i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.296 + 0.514i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.23 + 3.86i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.51T + 13T^{2} \) |
| 17 | \( 1 + (0.136 - 0.236i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.43 + 2.48i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.52 - 4.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.352T + 29T^{2} \) |
| 31 | \( 1 + (1.25 - 2.17i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.32 - 5.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 3.38T + 43T^{2} \) |
| 47 | \( 1 + (-6.21 - 10.7i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.66 + 9.80i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.02 - 6.97i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.36 - 2.36i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.93 - 5.08i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 + (5.55 - 9.62i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.58 + 9.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + (2.68 + 4.65i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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
−11.83399094579787224312406227773, −11.25464412325648791283986621158, −10.42594738686001698543445670507, −9.316898687775732819777068258768, −8.496230878752029202630775541281, −7.22550737500618431112605846810, −5.79672416169981861450977037760, −4.38581712729720874802127500090, −2.75295905682635696080946726110, −1.05382292861248251908236924670,
2.40491511976131716383366971973, 4.40244857594651719507545620912, 5.72347437717030477254089544245, 7.08073980977407271897873381889, 7.59681237389984299935567103570, 8.803964799284291286266773118524, 9.579718292438336800895305307580, 10.94575917784581109431827338093, 12.10154013492408082479071708865, 12.56915864459932393203259331739