L(s) = 1 | + (1.41 + 0.0428i)2-s + (0.831 + 0.222i)3-s + (1.99 + 0.121i)4-s + (−2.02 + 0.543i)5-s + (1.16 + 0.350i)6-s + (−2.63 + 0.233i)7-s + (2.81 + 0.256i)8-s + (−1.95 − 1.12i)9-s + (−2.88 + 0.681i)10-s + (1.03 − 3.85i)11-s + (1.63 + 0.545i)12-s + (−0.990 + 0.990i)13-s + (−3.73 + 0.216i)14-s − 1.80·15-s + (3.97 + 0.483i)16-s + (3.07 + 5.33i)17-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0303i)2-s + (0.480 + 0.128i)3-s + (0.998 + 0.0605i)4-s + (−0.906 + 0.242i)5-s + (0.476 + 0.143i)6-s + (−0.996 + 0.0881i)7-s + (0.995 + 0.0907i)8-s + (−0.652 − 0.376i)9-s + (−0.913 + 0.215i)10-s + (0.311 − 1.16i)11-s + (0.471 + 0.157i)12-s + (−0.274 + 0.274i)13-s + (−0.998 + 0.0579i)14-s − 0.466·15-s + (0.992 + 0.120i)16-s + (0.746 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67606 + 0.135520i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67606 + 0.135520i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 - 0.0428i)T \) |
| 7 | \( 1 + (2.63 - 0.233i)T \) |
good | 3 | \( 1 + (-0.831 - 0.222i)T + (2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (2.02 - 0.543i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.03 + 3.85i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (0.990 - 0.990i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.07 - 5.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.01 + 3.79i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-5.91 - 3.41i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.83 - 3.83i)T - 29iT^{2} \) |
| 31 | \( 1 + (2.05 + 3.55i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0740 + 0.0198i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 8.68iT - 41T^{2} \) |
| 43 | \( 1 + (-0.713 - 0.713i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.95 + 3.38i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.89 + 7.06i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.851 - 3.17i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (2.37 + 8.84i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (1.49 + 0.401i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 2.86iT - 71T^{2} \) |
| 73 | \( 1 + (-8.95 + 5.17i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.33 - 5.77i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.16 + 0.671i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 18.7T + 97T^{2} \) |
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
−13.63840408461535393344539152937, −12.77366899133456947417423073242, −11.64961496926830586694324748852, −10.93879478004517633909319485125, −9.323370781573329987069825100154, −8.076207917308130160078641360692, −6.78437692109769067320267831178, −5.68878777495598799961853375706, −3.75704324079825771597658890748, −3.12885775793798443163381927323,
2.73685861755943070441017125996, 3.99398810871188323944586383897, 5.39633796738455994524373375599, 6.99996653273255337129993257539, 7.79339512576288402726924570967, 9.369454653282047032140553325795, 10.66163278936194834266942047424, 12.00988336060552815401545267742, 12.48690574467080601496024624648, 13.61196765047574921778512831694