| L(s) = 1 | + 3-s + 2·7-s + 9-s + 6·13-s + 3·17-s + 2·19-s + 2·21-s + 3·23-s + 27-s + 4·29-s + 31-s − 10·37-s + 6·39-s + 3·41-s − 43-s + 7·47-s − 3·49-s + 3·51-s − 9·53-s + 2·57-s + 3·59-s + 2·61-s + 2·63-s + 11·67-s + 3·69-s − 14·71-s + 3·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.66·13-s + 0.727·17-s + 0.458·19-s + 0.436·21-s + 0.625·23-s + 0.192·27-s + 0.742·29-s + 0.179·31-s − 1.64·37-s + 0.960·39-s + 0.468·41-s − 0.152·43-s + 1.02·47-s − 3/7·49-s + 0.420·51-s − 1.23·53-s + 0.264·57-s + 0.390·59-s + 0.256·61-s + 0.251·63-s + 1.34·67-s + 0.361·69-s − 1.66·71-s + 0.337·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.502968501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.502968501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.97596564516633, −12.74392523119470, −11.95492519722534, −11.72734697742799, −11.11687567826363, −10.62835929132161, −10.37063907100588, −9.691377688762668, −9.066494989554411, −8.839602109821106, −8.225329797873011, −7.935304291089461, −7.448766717238535, −6.715392316930523, −6.439170326936102, −5.652428268521783, −5.274307919855262, −4.701434698606050, −4.085942727704586, −3.516956268970180, −3.174743895115277, −2.472517679797913, −1.681580132157306, −1.287624658754672, −0.6780201842895892,
0.6780201842895892, 1.287624658754672, 1.681580132157306, 2.472517679797913, 3.174743895115277, 3.516956268970180, 4.085942727704586, 4.701434698606050, 5.274307919855262, 5.652428268521783, 6.439170326936102, 6.715392316930523, 7.448766717238535, 7.935304291089461, 8.225329797873011, 8.839602109821106, 9.066494989554411, 9.691377688762668, 10.37063907100588, 10.62835929132161, 11.11687567826363, 11.72734697742799, 11.95492519722534, 12.74392523119470, 12.97596564516633
