L(s) = 1 | + (1.12 − 0.862i)2-s + (0.511 − 1.93i)4-s + (1.60 + 2.78i)5-s + (1.82 + 1.05i)7-s + (−1.09 − 2.60i)8-s + (4.20 + 1.73i)10-s + (−3.47 − 2.00i)11-s + (−0.341 + 0.197i)13-s + (2.94 − 0.392i)14-s + (−3.47 − 1.97i)16-s − 1.20i·17-s − 1.62·19-s + (6.21 − 1.68i)20-s + (−5.62 + 0.749i)22-s + (−2.74 − 4.75i)23-s + ⋯ |
L(s) = 1 | + (0.792 − 0.609i)2-s + (0.255 − 0.966i)4-s + (0.719 + 1.24i)5-s + (0.688 + 0.397i)7-s + (−0.386 − 0.922i)8-s + (1.33 + 0.548i)10-s + (−1.04 − 0.605i)11-s + (−0.0948 + 0.0547i)13-s + (0.788 − 0.105i)14-s + (−0.869 − 0.494i)16-s − 0.292i·17-s − 0.372·19-s + (1.38 − 0.376i)20-s + (−1.20 + 0.159i)22-s + (−0.572 − 0.990i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.89470 - 0.529346i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.89470 - 0.529346i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.12 + 0.862i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.60 - 2.78i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.82 - 1.05i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.47 + 2.00i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.341 - 0.197i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.20iT - 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 + (2.74 + 4.75i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.95 - 5.12i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.34 + 1.93i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 + (-1.23 + 0.715i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.21 - 2.10i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.792 - 1.37i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 + (-2.29 + 1.32i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.18 + 4.72i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.60 + 4.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 - 9.49T + 73T^{2} \) |
| 79 | \( 1 + (-1.53 - 0.886i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.30 - 0.755i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 11.2iT - 89T^{2} \) |
| 97 | \( 1 + (-5.84 + 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.19883677807832864871933717886, −11.14516344180139896258843120480, −10.59404400242735637965169848793, −9.745220451169809004624846941034, −8.255691767984928416600951979076, −6.79109683545752707774743806822, −5.87923603800714144416519818603, −4.83309964051970877593037703554, −3.11087842226862977687247479677, −2.17788314410282264070507524753,
2.08784552186305106219218933182, 4.15779471407644360437760518384, 5.10025993369227482620920628679, 5.85746704742874363973750593970, 7.43370933310331235205427069731, 8.184929101223090810046542315474, 9.280554339616210428563314160049, 10.51526722722042845185272523773, 11.78448320695217600722749541162, 12.72535648296736909160206129784