L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 4·11-s − 4·13-s − 15-s + 2·17-s + 8·19-s − 2·21-s + 8·23-s + 25-s − 27-s + 4·29-s + 31-s − 4·33-s + 2·35-s − 12·37-s + 4·39-s + 10·41-s − 8·43-s + 45-s + 4·47-s − 3·49-s − 2·51-s + 6·53-s + 4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s − 0.258·15-s + 0.485·17-s + 1.83·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.179·31-s − 0.696·33-s + 0.338·35-s − 1.97·37-s + 0.640·39-s + 1.56·41-s − 1.21·43-s + 0.149·45-s + 0.583·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.483967128\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.483967128\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.73607329399803119395495322518, −7.04545572599436585427854318592, −6.69974762581254571811671183419, −5.53697114966168480167946148377, −5.21425268691975651513929567957, −4.55910921470387361475142056831, −3.54592134070390671333408463172, −2.69356100104222580603901645737, −1.52843138728323757710674036806, −0.917309887421169471303075548523,
0.917309887421169471303075548523, 1.52843138728323757710674036806, 2.69356100104222580603901645737, 3.54592134070390671333408463172, 4.55910921470387361475142056831, 5.21425268691975651513929567957, 5.53697114966168480167946148377, 6.69974762581254571811671183419, 7.04545572599436585427854318592, 7.73607329399803119395495322518