| L(s) = 1 | + 0.417·5-s − 2.64i·7-s − 18.4i·11-s − 1.16·13-s − 0.417·17-s + 21.1i·19-s − 16.9i·23-s − 24.8·25-s − 4.33·29-s + 20.7i·31-s − 1.10i·35-s − 61.1·37-s + 9.07·41-s + 7.30i·43-s − 19.3i·47-s + ⋯ |
| L(s) = 1 | + 0.0834·5-s − 0.377i·7-s − 1.67i·11-s − 0.0896·13-s − 0.0245·17-s + 1.11i·19-s − 0.738i·23-s − 0.993·25-s − 0.149·29-s + 0.670i·31-s − 0.0315i·35-s − 1.65·37-s + 0.221·41-s + 0.169i·43-s − 0.411i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8106490278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8106490278\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 - 0.417T + 25T^{2} \) |
| 11 | \( 1 + 18.4iT - 121T^{2} \) |
| 13 | \( 1 + 1.16T + 169T^{2} \) |
| 17 | \( 1 + 0.417T + 289T^{2} \) |
| 19 | \( 1 - 21.1iT - 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 4.33T + 841T^{2} \) |
| 31 | \( 1 - 20.7iT - 961T^{2} \) |
| 37 | \( 1 + 61.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 9.07T + 1.68e3T^{2} \) |
| 43 | \( 1 - 7.30iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 19.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 92.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 99.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 78.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 77.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 43.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 53.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 74.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 81.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 30.1T + 9.40e3T^{2} \) |
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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
−9.437250473806668342833509032654, −8.448578477898126094406512025439, −7.965976390023830101725925106450, −6.78449021887202028778772381892, −6.00441550857166226709966488223, −5.17034619869508819068973487653, −3.89196201388116045559357443152, −3.14998162834850840718318484436, −1.66152854141983946030452881059, −0.24253866726714991478179809896,
1.66888496827976211595510568130, 2.63430812296814215996378862597, 4.00039556568896174478415418899, 4.89453473107546022578045190688, 5.76103419621394904113724271181, 6.92290593930275799970407245222, 7.45184167180686993396239774038, 8.509712944347277815436904762428, 9.484864536852351974253553660034, 9.866302195686111785638235081543
