L(s) = 1 | + (−0.671 − 1.24i)2-s + (1.72 + 0.162i)3-s + (−1.09 + 1.67i)4-s + (0.592 − 2.21i)5-s + (−0.956 − 2.25i)6-s + (−2.67 − 1.54i)7-s + (2.81 + 0.241i)8-s + (2.94 + 0.559i)9-s + (−3.14 + 0.748i)10-s + (3.42 − 0.918i)11-s + (−2.16 + 2.70i)12-s + (1.40 + 0.375i)13-s + (−0.124 + 4.35i)14-s + (1.38 − 3.71i)15-s + (−1.59 − 3.66i)16-s − 1.69·17-s + ⋯ |
L(s) = 1 | + (−0.475 − 0.879i)2-s + (0.995 + 0.0936i)3-s + (−0.548 + 0.836i)4-s + (0.264 − 0.988i)5-s + (−0.390 − 0.920i)6-s + (−1.00 − 0.582i)7-s + (0.996 + 0.0852i)8-s + (0.982 + 0.186i)9-s + (−0.996 + 0.236i)10-s + (1.03 − 0.277i)11-s + (−0.624 + 0.781i)12-s + (0.389 + 0.104i)13-s + (−0.0331 + 1.16i)14-s + (0.356 − 0.959i)15-s + (−0.398 − 0.917i)16-s − 0.411·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.863692 - 0.721774i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.863692 - 0.721774i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.671 + 1.24i)T \) |
| 3 | \( 1 + (-1.72 - 0.162i)T \) |
good | 5 | \( 1 + (-0.592 + 2.21i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (2.67 + 1.54i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.42 + 0.918i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.40 - 0.375i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + 1.69T + 17T^{2} \) |
| 19 | \( 1 + (5.41 - 5.41i)T - 19iT^{2} \) |
| 23 | \( 1 + (3.69 - 2.13i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.147 - 0.550i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-3.59 - 6.22i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.59 - 2.59i)T + 37iT^{2} \) |
| 41 | \( 1 + (-8.14 + 4.70i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (10.5 - 2.81i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (0.322 - 0.558i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.59 + 7.59i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.55 - 5.82i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.04 - 3.88i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-4.02 - 1.07i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 4.81iT - 71T^{2} \) |
| 73 | \( 1 - 0.0254iT - 73T^{2} \) |
| 79 | \( 1 + (-7.90 + 13.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.50 + 9.35i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 1.29iT - 89T^{2} \) |
| 97 | \( 1 + (-4.31 + 7.46i)T + (-48.5 - 84.0i)T^{2} \) |
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
−12.92577110311065039101437131082, −12.06675875174194226083511070897, −10.52896302807167478283185197020, −9.675949083009300676448065860803, −8.903005234970544550539242645627, −8.128694726836196599087043888188, −6.57914357881792951950614812912, −4.34766978269543131371731084798, −3.45295181548362768904442104336, −1.56013491937843888351242037489,
2.47802810580879995628389108282, 4.17412272119277732118798468384, 6.43968322223574698707253312758, 6.63103173424708491849577937748, 8.133906781306344398324196773834, 9.185107455875748820604708821863, 9.780355265875620947001349221671, 10.96013690671478601755999690867, 12.71494778950219406582122366989, 13.61471135460594049323476330854