| L(s) = 1 | + 2·5-s + 4·7-s − 2·11-s + 8·19-s + 3·25-s − 4·29-s + 4·31-s + 8·35-s + 4·37-s − 4·41-s + 4·43-s + 8·47-s − 4·53-s − 4·55-s + 12·59-s + 12·61-s + 4·71-s − 8·77-s + 16·79-s + 12·83-s − 4·89-s + 16·95-s + 12·97-s − 20·101-s + 12·107-s − 4·109-s − 4·113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 0.603·11-s + 1.83·19-s + 3/5·25-s − 0.742·29-s + 0.718·31-s + 1.35·35-s + 0.657·37-s − 0.624·41-s + 0.609·43-s + 1.16·47-s − 0.549·53-s − 0.539·55-s + 1.56·59-s + 1.53·61-s + 0.474·71-s − 0.911·77-s + 1.80·79-s + 1.31·83-s − 0.423·89-s + 1.64·95-s + 1.21·97-s − 1.99·101-s + 1.16·107-s − 0.383·109-s − 0.376·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.392199739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.392199739\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 184 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.86742359342845289971225402962, −7.84788073694050585644894833015, −7.35168742222963915324465063029, −7.08494715646822316997389985779, −6.50654914424554425856181397563, −6.43249903475446675747119669406, −5.70387726442982119002732516346, −5.52082698066106743555588560117, −5.18362500483067615029293687711, −5.12527526823527625714200989903, −4.45775520727538247925520386588, −4.32246301787294433728658447599, −3.53046985465691196970212048177, −3.45290856677079954675668782919, −2.62095387748962369119489522305, −2.56220393342996253562323725020, −1.86125905996384875704005040948, −1.70853975197037744690635787908, −0.909602297106672650127731883100, −0.73290838982016040638422844127,
0.73290838982016040638422844127, 0.909602297106672650127731883100, 1.70853975197037744690635787908, 1.86125905996384875704005040948, 2.56220393342996253562323725020, 2.62095387748962369119489522305, 3.45290856677079954675668782919, 3.53046985465691196970212048177, 4.32246301787294433728658447599, 4.45775520727538247925520386588, 5.12527526823527625714200989903, 5.18362500483067615029293687711, 5.52082698066106743555588560117, 5.70387726442982119002732516346, 6.43249903475446675747119669406, 6.50654914424554425856181397563, 7.08494715646822316997389985779, 7.35168742222963915324465063029, 7.84788073694050585644894833015, 7.86742359342845289971225402962
