L(s) = 1 | + 0.732·3-s − 1.26i·7-s − 2.46·9-s − 3.46i·11-s − 3.46·13-s − 3.46i·17-s + 2i·19-s − 0.928i·21-s + 8.19i·23-s − 4·27-s − 9.46·31-s − 2.53i·33-s − 6·37-s − 2.53·39-s + 2.53·41-s + ⋯ |
L(s) = 1 | + 0.422·3-s − 0.479i·7-s − 0.821·9-s − 1.04i·11-s − 0.960·13-s − 0.840i·17-s + 0.458i·19-s − 0.202i·21-s + 1.70i·23-s − 0.769·27-s − 1.69·31-s − 0.441i·33-s − 0.986·37-s − 0.406·39-s + 0.396·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4151260200\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4151260200\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 1.26iT - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 8.19iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8.19iT - 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 + 14.3iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4.73T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 6.39iT - 97T^{2} \) |
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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
−9.244670171831600163752803198162, −8.100815152269345991116377244344, −7.63791194931499478866722655297, −6.69960107900960116719417169282, −5.62580782303062663980840493457, −5.03993600287336669442377053856, −3.61930928510390447413421861623, −3.11745652410939262085376224231, −1.81698050031540470105310980378, −0.13703573424083539679759682059,
1.98484866011011060993434982237, 2.66176257245725524621528595562, 3.83556444474130628999796501594, 4.87097588614781965947943590383, 5.60871129735776259200355235388, 6.68929986677692118510140706360, 7.37194253983830424493626792822, 8.400202751353899356118586467515, 8.842099123465698324029027854075, 9.720605872197416620652651035440