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 | \( 1 + (-4.31 - 7.46i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.61471135460594049323476330854, −12.71494778950219406582122366989, −10.96013690671478601755999690867, −9.780355265875620947001349221671, −9.185107455875748820604708821863, −8.133906781306344398324196773834, −6.63103173424708491849577937748, −6.43968322223574698707253312758, −4.17412272119277732118798468384, −2.47802810580879995628389108282,
1.56013491937843888351242037489, 3.45295181548362768904442104336, 4.34766978269543131371731084798, 6.57914357881792951950614812912, 8.128694726836196599087043888188, 8.903005234970544550539242645627, 9.675949083009300676448065860803, 10.52896302807167478283185197020, 12.06675875174194226083511070897, 12.92577110311065039101437131082