| L(s) = 1 | − 3-s + 5-s + 9-s − 3·11-s + 4·13-s − 15-s − 4·17-s + 19-s − 4·23-s − 4·25-s − 27-s + 7·29-s + 7·31-s + 3·33-s + 4·37-s − 4·39-s + 12·41-s + 6·43-s + 45-s + 10·47-s + 4·51-s − 13·53-s − 3·55-s − 57-s + 59-s − 14·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 0.258·15-s − 0.970·17-s + 0.229·19-s − 0.834·23-s − 4/5·25-s − 0.192·27-s + 1.29·29-s + 1.25·31-s + 0.522·33-s + 0.657·37-s − 0.640·39-s + 1.87·41-s + 0.914·43-s + 0.149·45-s + 1.45·47-s + 0.560·51-s − 1.78·53-s − 0.404·55-s − 0.132·57-s + 0.130·59-s − 1.79·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.090495169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.090495169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.04258400198403, −12.79319865138610, −12.22231379886279, −11.78402569521865, −11.06152605342940, −10.92742234485150, −10.45620075465152, −9.846026893926075, −9.475325014828161, −8.938407476320776, −8.278045188756239, −7.900578690158745, −7.483103232180626, −6.634474920569099, −6.244660589149925, −5.969992635905328, −5.409953252599931, −4.765079497636185, −4.226901771950501, −3.903643276595885, −2.808980374751035, −2.605087759120425, −1.815637591175357, −1.089135003933822, −0.4770759932522439,
0.4770759932522439, 1.089135003933822, 1.815637591175357, 2.605087759120425, 2.808980374751035, 3.903643276595885, 4.226901771950501, 4.765079497636185, 5.409953252599931, 5.969992635905328, 6.244660589149925, 6.634474920569099, 7.483103232180626, 7.900578690158745, 8.278045188756239, 8.938407476320776, 9.475325014828161, 9.846026893926075, 10.45620075465152, 10.92742234485150, 11.06152605342940, 11.78402569521865, 12.22231379886279, 12.79319865138610, 13.04258400198403
