L(s) = 1 | + (1.75 − 1.75i)2-s + (−1.71 − 0.230i)3-s − 4.14i·4-s + (−0.542 − 0.145i)5-s + (−3.41 + 2.60i)6-s + (0.541 + 0.144i)7-s + (−3.76 − 3.76i)8-s + (2.89 + 0.790i)9-s + (−1.20 + 0.695i)10-s + (3.72 + 3.72i)11-s + (−0.954 + 7.11i)12-s + (−1.98 − 3.00i)13-s + (1.20 − 0.694i)14-s + (0.897 + 0.374i)15-s − 4.89·16-s + (−1.39 + 2.42i)17-s + ⋯ |
L(s) = 1 | + (1.23 − 1.23i)2-s + (−0.991 − 0.132i)3-s − 2.07i·4-s + (−0.242 − 0.0649i)5-s + (−1.39 + 1.06i)6-s + (0.204 + 0.0548i)7-s + (−1.33 − 1.33i)8-s + (0.964 + 0.263i)9-s + (−0.381 + 0.220i)10-s + (1.12 + 1.12i)11-s + (−0.275 + 2.05i)12-s + (−0.551 − 0.833i)13-s + (0.321 − 0.185i)14-s + (0.231 + 0.0966i)15-s − 1.22·16-s + (−0.339 + 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.829483 - 1.17506i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.829483 - 1.17506i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.71 + 0.230i)T \) |
| 13 | \( 1 + (1.98 + 3.00i)T \) |
good | 2 | \( 1 + (-1.75 + 1.75i)T - 2iT^{2} \) |
| 5 | \( 1 + (0.542 + 0.145i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-0.541 - 0.144i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.72 - 3.72i)T + 11iT^{2} \) |
| 17 | \( 1 + (1.39 - 2.42i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.31 + 0.888i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (3.77 - 6.54i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.895iT - 29T^{2} \) |
| 31 | \( 1 + (1.57 - 5.88i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (0.0565 + 0.0151i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.50 + 5.62i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-9.03 + 5.21i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (10.6 - 2.84i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 - 5.68iT - 53T^{2} \) |
| 59 | \( 1 + (-3.61 - 3.61i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.08 - 3.61i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.36 - 1.70i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.573 + 2.14i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-2.68 + 2.68i)T - 73iT^{2} \) |
| 79 | \( 1 + (-2.61 + 4.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.0375 - 0.139i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-2.14 + 7.99i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (0.238 - 0.888i)T + (-84.0 - 48.5i)T^{2} \) |
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
−12.84028501432569596028431181534, −12.07320427631109213944028338973, −11.62043384785452670697109552374, −10.45966050125281435103924950289, −9.645295919092598138348551977908, −7.39077814545992625511259119958, −5.94092281836995385137538293099, −4.88298267584568633370190108188, −3.82914969531675339274106637244, −1.72032583047790015929015760158,
3.81745336379555033649068493798, 4.80211670789593247920552744434, 6.04512037318534558053831305780, 6.74886949064069226863534211235, 7.936052376637265196644288053581, 9.511458197993654267626450501499, 11.37941894533098680640068030490, 11.83345319635196840458947689197, 12.99457272385663898374086719555, 14.08610786100480598136801671420