L(s) = 1 | + 5-s − 6·13-s − 7·17-s + 4·19-s + 3·23-s + 25-s − 4·29-s − 3·31-s + 2·41-s − 4·43-s − 7·47-s − 7·49-s + 5·53-s + 4·59-s + 61-s − 6·65-s − 2·67-s − 10·71-s − 4·73-s + 5·79-s + 12·83-s − 7·85-s + 4·95-s + 6·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.66·13-s − 1.69·17-s + 0.917·19-s + 0.625·23-s + 1/5·25-s − 0.742·29-s − 0.538·31-s + 0.312·41-s − 0.609·43-s − 1.02·47-s − 49-s + 0.686·53-s + 0.520·59-s + 0.128·61-s − 0.744·65-s − 0.244·67-s − 1.18·71-s − 0.468·73-s + 0.562·79-s + 1.31·83-s − 0.759·85-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.036308191\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.036308191\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.72543621200102, −13.45445005568990, −12.95552263321042, −12.50898541069096, −11.89406075704013, −11.39788282357648, −11.03653780851827, −10.31475769059574, −9.923933502718397, −9.293439179987602, −9.123170919282147, −8.410447218150696, −7.726355863502558, −7.242906095752530, −6.833109103869335, −6.279795054286720, −5.564426230802487, −4.960924498577047, −4.752443510267637, −3.933752968017773, −3.216760227281059, −2.559934877321254, −2.091578143303468, −1.391698062584764, −0.3140407624396482,
0.3140407624396482, 1.391698062584764, 2.091578143303468, 2.559934877321254, 3.216760227281059, 3.933752968017773, 4.752443510267637, 4.960924498577047, 5.564426230802487, 6.279795054286720, 6.833109103869335, 7.242906095752530, 7.726355863502558, 8.410447218150696, 9.123170919282147, 9.293439179987602, 9.923933502718397, 10.31475769059574, 11.03653780851827, 11.39788282357648, 11.89406075704013, 12.50898541069096, 12.95552263321042, 13.45445005568990, 13.72543621200102