L(s) = 1 | + (0.454 + 0.262i)2-s + (−1.11 − 1.32i)3-s + (−0.862 − 1.49i)4-s + (−2.14 + 0.616i)5-s + (−0.159 − 0.894i)6-s + (0.133 + 0.0768i)7-s − 1.95i·8-s + (−0.506 + 2.95i)9-s + (−1.13 − 0.283i)10-s + (0.5 − 0.866i)11-s + (−1.01 + 2.80i)12-s + (−3.09 + 1.78i)13-s + (0.0403 + 0.0698i)14-s + (3.21 + 2.15i)15-s + (−1.21 + 2.09i)16-s + 4.15i·17-s + ⋯ |
L(s) = 1 | + (0.321 + 0.185i)2-s + (−0.644 − 0.764i)3-s + (−0.431 − 0.746i)4-s + (−0.961 + 0.275i)5-s + (−0.0653 − 0.365i)6-s + (0.0503 + 0.0290i)7-s − 0.690i·8-s + (−0.168 + 0.985i)9-s + (−0.359 − 0.0898i)10-s + (0.150 − 0.261i)11-s + (−0.292 + 0.811i)12-s + (−0.857 + 0.494i)13-s + (0.0107 + 0.0186i)14-s + (0.830 + 0.557i)15-s + (−0.303 + 0.524i)16-s + 1.00i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.101250 + 0.162890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.101250 + 0.162890i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.11 + 1.32i)T \) |
| 5 | \( 1 + (2.14 - 0.616i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.454 - 0.262i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.133 - 0.0768i)T + (3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (3.09 - 1.78i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.15iT - 17T^{2} \) |
| 19 | \( 1 - 4.91T + 19T^{2} \) |
| 23 | \( 1 + (4.22 - 2.43i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.11 - 8.85i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.56 + 7.90i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.68iT - 37T^{2} \) |
| 41 | \( 1 + (-3.30 - 5.72i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.96 + 2.86i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.52 + 3.18i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9.44iT - 53T^{2} \) |
| 59 | \( 1 + (-3.08 - 5.34i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.845 - 1.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.12 - 3.53i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 16.5iT - 73T^{2} \) |
| 79 | \( 1 + (1.18 - 2.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.3 - 7.13i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 7.56T + 89T^{2} \) |
| 97 | \( 1 + (0.350 + 0.202i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.44234203854724531369010175508, −10.49008429640017787961451659726, −9.569788981025790720975441549955, −8.322686399312601515766166306077, −7.40869864662000940744546341042, −6.62417648157536934698366370660, −5.63479943024967846840765009917, −4.76624505733883706914973947965, −3.57709887275798112511632128813, −1.61361899800584084118710452311,
0.11501324619519461440406518861, 2.97830535221872667181295925439, 3.97900954269131116319317585751, 4.74339225888550616643814118862, 5.55139451089822926243959317938, 7.22077492135836057561991263241, 7.84767336322038503263782007378, 9.051394279107427371538253236461, 9.692117316538243318689392993588, 10.90386982119545049674236786063