L(s) = 1 | + (0.686 + 0.396i)5-s + (2.35 − 1.35i)7-s + (−1.71 − 2.96i)11-s + (1.68 − 2.92i)13-s + 2.52i·17-s − 2.20i·19-s + (1.07 − 1.86i)23-s + (−2.18 − 3.78i)25-s + (−0.686 + 0.396i)29-s + (−1.47 − 0.852i)31-s + 2.15·35-s − 4.74·37-s + (−0.127 − 0.0737i)41-s + (6.01 − 3.47i)43-s + (−5.77 − 10.0i)47-s + ⋯ |
L(s) = 1 | + (0.306 + 0.177i)5-s + (0.888 − 0.513i)7-s + (−0.516 − 0.894i)11-s + (0.467 − 0.809i)13-s + 0.612i·17-s − 0.506i·19-s + (0.224 − 0.388i)23-s + (−0.437 − 0.757i)25-s + (−0.127 + 0.0735i)29-s + (−0.265 − 0.153i)31-s + 0.363·35-s − 0.780·37-s + (−0.0199 − 0.0115i)41-s + (0.917 − 0.529i)43-s + (−0.842 − 1.45i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.801299745\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.801299745\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.686 - 0.396i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.35 + 1.35i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.71 + 2.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.68 + 2.92i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2.52iT - 17T^{2} \) |
| 19 | \( 1 + 2.20iT - 19T^{2} \) |
| 23 | \( 1 + (-1.07 + 1.86i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.686 - 0.396i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.47 + 0.852i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 + (0.127 + 0.0737i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.01 + 3.47i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.77 + 10.0i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.51iT - 53T^{2} \) |
| 59 | \( 1 + (-2.58 + 4.48i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.68 - 2.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.01 - 3.47i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.75T + 71T^{2} \) |
| 73 | \( 1 + 2.37T + 73T^{2} \) |
| 79 | \( 1 + (-8.80 + 5.08i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.62 + 6.28i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5.34iT - 89T^{2} \) |
| 97 | \( 1 + (-6.24 - 10.8i)T + (-48.5 + 84.0i)T^{2} \) |
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
−9.002578163554806812512325912979, −8.297480724698401473690940395334, −7.75937297135682242677336922491, −6.75528532295188936819756578867, −5.84696153217729026206543557321, −5.17292165566225755456219600537, −4.12968229475188219027901664176, −3.17131780330014694596399757639, −2.02905897925084089394173579564, −0.69339152863933710561095272778,
1.51178308547729397445461668758, 2.26608696023373612616145897765, 3.60771864916944807005549439080, 4.73536988650635434784332675802, 5.26312458732632940360036231402, 6.20545554118235489697857179889, 7.21894148573328772295929386997, 7.87630068757940461021273780101, 8.752329692428958034939566585207, 9.442833069369849731397546067657