L(s) = 1 | + (0.488 − 0.488i)3-s − 4.71·7-s + 2.52i·9-s + (3.91 − 3.91i)11-s + (−0.0878 + 0.0878i)13-s + 4.67i·17-s + (1.81 + 1.81i)19-s + (−2.30 + 2.30i)21-s − 1.63·23-s + (2.69 + 2.69i)27-s + (−3.26 − 3.26i)29-s + 2.12·31-s − 3.82i·33-s + (3.97 + 3.97i)37-s + 0.0858i·39-s + ⋯ |
L(s) = 1 | + (0.282 − 0.282i)3-s − 1.78·7-s + 0.840i·9-s + (1.17 − 1.17i)11-s + (−0.0243 + 0.0243i)13-s + 1.13i·17-s + (0.415 + 0.415i)19-s + (−0.502 + 0.502i)21-s − 0.339·23-s + (0.519 + 0.519i)27-s + (−0.606 − 0.606i)29-s + 0.382·31-s − 0.665i·33-s + (0.653 + 0.653i)37-s + 0.0137i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.357772261\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.357772261\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.488 + 0.488i)T - 3iT^{2} \) |
| 7 | \( 1 + 4.71T + 7T^{2} \) |
| 11 | \( 1 + (-3.91 + 3.91i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.0878 - 0.0878i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.67iT - 17T^{2} \) |
| 19 | \( 1 + (-1.81 - 1.81i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 + (3.26 + 3.26i)T + 29iT^{2} \) |
| 31 | \( 1 - 2.12T + 31T^{2} \) |
| 37 | \( 1 + (-3.97 - 3.97i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.25iT - 41T^{2} \) |
| 43 | \( 1 + (-2.27 - 2.27i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.06iT - 47T^{2} \) |
| 53 | \( 1 + (-5.03 - 5.03i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.16 - 5.16i)T - 59iT^{2} \) |
| 61 | \( 1 + (-7.12 - 7.12i)T + 61iT^{2} \) |
| 67 | \( 1 + (7.49 - 7.49i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.54iT - 71T^{2} \) |
| 73 | \( 1 - 8.30T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + (-1.16 + 1.16i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.24iT - 89T^{2} \) |
| 97 | \( 1 + 13.9iT - 97T^{2} \) |
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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
−9.535188694189700429620423824498, −8.722683415880119426432127747015, −8.019106642858102778995174471957, −7.06387666628999102618541251206, −6.12072616558178745350917673226, −5.90165860322780381788152821844, −4.26366175228022466339278895014, −3.47079538777307163282359323344, −2.65265568736742455409684922352, −1.18776013242185260190915097868,
0.57041123053864873518910872855, 2.33445762673132776451860395137, 3.44747510774273819988696551347, 3.92877871876725428828165962600, 5.12602010536482251791613243598, 6.31995848728357088430580797580, 6.79250017731106240959334674714, 7.45523130370712915303177344544, 8.940689834973644084372778152021, 9.461264393612599397073588738297