| L(s) = 1 | − 5-s + 7-s + 4·11-s + 6·13-s + 4·17-s + 2·19-s + 8·23-s + 25-s + 6·29-s − 4·31-s − 35-s − 8·37-s − 2·41-s − 6·43-s − 6·47-s + 49-s + 2·53-s − 4·55-s + 4·59-s − 6·65-s − 14·67-s − 2·71-s + 6·73-s + 4·77-s − 8·79-s − 8·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.970·17-s + 0.458·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.169·35-s − 1.31·37-s − 0.312·41-s − 0.914·43-s − 0.875·47-s + 1/7·49-s + 0.274·53-s − 0.539·55-s + 0.520·59-s − 0.744·65-s − 1.71·67-s − 0.237·71-s + 0.702·73-s + 0.455·77-s − 0.900·79-s − 0.878·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.520220825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520220825\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.512167260296311137679267860154, −7.44635719600484315392126468392, −6.84398788856244479184179944781, −6.13668828927939727244945936087, −5.28943645399572705078113480564, −4.52924946627319521544994174747, −3.52633327642886197182115880849, −3.23749852648338313501483929596, −1.58884488351361440165452121229, −0.989078308654992976657654224665,
0.989078308654992976657654224665, 1.58884488351361440165452121229, 3.23749852648338313501483929596, 3.52633327642886197182115880849, 4.52924946627319521544994174747, 5.28943645399572705078113480564, 6.13668828927939727244945936087, 6.84398788856244479184179944781, 7.44635719600484315392126468392, 8.512167260296311137679267860154
