| L(s) = 1 | − 10·7-s − 6·9-s − 12·11-s + 60·23-s − 12·29-s − 20·37-s − 20·43-s + 51·49-s − 180·53-s + 60·63-s + 140·67-s + 84·71-s + 120·77-s + 148·79-s − 45·81-s + 72·99-s + 300·107-s − 172·109-s − 180·113-s − 134·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1.42·7-s − 2/3·9-s − 1.09·11-s + 2.60·23-s − 0.413·29-s − 0.540·37-s − 0.465·43-s + 1.04·49-s − 3.39·53-s + 0.952·63-s + 2.08·67-s + 1.18·71-s + 1.55·77-s + 1.87·79-s − 5/9·81-s + 8/11·99-s + 2.80·107-s − 1.57·109-s − 1.59·113-s − 1.10·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7206320039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7206320039\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 10 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 p T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 314 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 194 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 122 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 962 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4034 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 90 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6362 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6842 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 958 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 9722 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 5758 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12674 T^{2} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41420024668871922543861910060, −10.08640556090955697077195067890, −9.364547582152469431964703419546, −9.363561929688620917098301921427, −8.881156682705261317730567432822, −8.280560916387947572423297840105, −7.85655169522380451117815198673, −7.45930217300974876017871829678, −6.72101177506260987954449214119, −6.61187312938593199805243495832, −6.13720540228307089824412761221, −5.27735564687474710607835416155, −5.21426099884037038524693684078, −4.68977052231077674445650292227, −3.59388808893838526577584980523, −3.44493452046416911712917310800, −2.78343822597224295464291830564, −2.42524437496801564336014123283, −1.28970954383693666403895442675, −0.31372547562723234750719186747,
0.31372547562723234750719186747, 1.28970954383693666403895442675, 2.42524437496801564336014123283, 2.78343822597224295464291830564, 3.44493452046416911712917310800, 3.59388808893838526577584980523, 4.68977052231077674445650292227, 5.21426099884037038524693684078, 5.27735564687474710607835416155, 6.13720540228307089824412761221, 6.61187312938593199805243495832, 6.72101177506260987954449214119, 7.45930217300974876017871829678, 7.85655169522380451117815198673, 8.280560916387947572423297840105, 8.881156682705261317730567432822, 9.363561929688620917098301921427, 9.364547582152469431964703419546, 10.08640556090955697077195067890, 10.41420024668871922543861910060
