| L(s) = 1 | − 6·11-s − 3·13-s + 4·17-s + 19-s + 4·23-s + 8·29-s + 31-s − 7·37-s − 6·41-s + 43-s + 2·47-s + 4·53-s + 8·59-s + 14·61-s + 7·67-s + 6·71-s + 73-s + 79-s + 2·83-s − 12·89-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.80·11-s − 0.832·13-s + 0.970·17-s + 0.229·19-s + 0.834·23-s + 1.48·29-s + 0.179·31-s − 1.15·37-s − 0.937·41-s + 0.152·43-s + 0.291·47-s + 0.549·53-s + 1.04·59-s + 1.79·61-s + 0.855·67-s + 0.712·71-s + 0.117·73-s + 0.112·79-s + 0.219·83-s − 1.27·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.921011029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921011029\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.21165982359859, −12.64809987053344, −12.25929172789211, −11.93436046370076, −11.22790857406606, −10.75415697019916, −10.28358889350982, −9.907583072837853, −9.620982946905458, −8.696078632197337, −8.349070936570486, −7.986205294120649, −7.324199467554373, −6.995926506461607, −6.487225746387115, −5.566104791681006, −5.220376284362238, −5.078286398748597, −4.278066096422936, −3.574539017575065, −2.940669477865479, −2.601730120839936, −1.987938736969190, −1.048166994412927, −0.4444708679029590,
0.4444708679029590, 1.048166994412927, 1.987938736969190, 2.601730120839936, 2.940669477865479, 3.574539017575065, 4.278066096422936, 5.078286398748597, 5.220376284362238, 5.566104791681006, 6.487225746387115, 6.995926506461607, 7.324199467554373, 7.986205294120649, 8.349070936570486, 8.696078632197337, 9.620982946905458, 9.907583072837853, 10.28358889350982, 10.75415697019916, 11.22790857406606, 11.93436046370076, 12.25929172789211, 12.64809987053344, 13.21165982359859
