L(s) = 1 | − 3-s − 7-s + 9-s + 11-s − 7·13-s + 2·17-s + 5·19-s + 21-s + 8·23-s − 27-s + 3·29-s − 8·31-s − 33-s + 4·37-s + 7·39-s + 4·41-s − 43-s + 7·47-s − 6·49-s − 2·51-s − 4·53-s − 5·57-s − 12·59-s − 63-s − 10·67-s − 8·69-s + 6·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.94·13-s + 0.485·17-s + 1.14·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s + 0.557·29-s − 1.43·31-s − 0.174·33-s + 0.657·37-s + 1.12·39-s + 0.624·41-s − 0.152·43-s + 1.02·47-s − 6/7·49-s − 0.280·51-s − 0.549·53-s − 0.662·57-s − 1.56·59-s − 0.125·63-s − 1.22·67-s − 0.963·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.443165679\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.443165679\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.55626760776294, −14.05897353505598, −13.39453729216768, −12.79358797358747, −12.35081567825926, −12.10262343526464, −11.38915277843623, −10.92812125776953, −10.40822943937097, −9.638481122411409, −9.464747715258403, −9.024093098964219, −7.946295702231724, −7.577434557156944, −7.026956234880769, −6.664042671704413, −5.779772105447442, −5.356724671028714, −4.804717186073586, −4.321726483911904, −3.291633743432308, −2.967460044378306, −2.114602737561493, −1.217856440499518, −0.4730034440505438,
0.4730034440505438, 1.217856440499518, 2.114602737561493, 2.967460044378306, 3.291633743432308, 4.321726483911904, 4.804717186073586, 5.356724671028714, 5.779772105447442, 6.664042671704413, 7.026956234880769, 7.577434557156944, 7.946295702231724, 9.024093098964219, 9.464747715258403, 9.638481122411409, 10.40822943937097, 10.92812125776953, 11.38915277843623, 12.10262343526464, 12.35081567825926, 12.79358797358747, 13.39453729216768, 14.05897353505598, 14.55626760776294