| L(s) = 1 | − 2·3-s − 4·5-s + 3·9-s + 4·11-s + 8·15-s − 12·17-s + 8·19-s + 4·23-s + 4·25-s − 4·27-s − 8·33-s + 8·37-s − 12·41-s − 12·45-s + 8·47-s + 24·51-s − 4·53-s − 16·55-s − 16·57-s − 8·61-s − 8·69-s + 4·71-s − 24·73-s − 8·75-s − 16·79-s + 5·81-s − 8·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 9-s + 1.20·11-s + 2.06·15-s − 2.91·17-s + 1.83·19-s + 0.834·23-s + 4/5·25-s − 0.769·27-s − 1.39·33-s + 1.31·37-s − 1.87·41-s − 1.78·45-s + 1.16·47-s + 3.36·51-s − 0.549·53-s − 2.15·55-s − 2.11·57-s − 1.02·61-s − 0.963·69-s + 0.474·71-s − 2.80·73-s − 0.923·75-s − 1.80·79-s + 5/9·81-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 24 T + 272 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.666309315043024180010044936232, −8.633896624027348567285064966310, −7.72806953602727379737167512769, −7.70170300688637950129612773536, −7.10276400475580769464366541150, −6.96984238991792902771881241245, −6.40017318611099806179872290760, −6.34517138040762474760387129009, −5.48100953375930169935458358780, −5.32470870015525549358797508304, −4.54388982970115817675053300559, −4.45508437457356945914192701442, −3.98060992217025302004451488127, −3.79029730786822254235881330119, −2.93187929445904833014945174614, −2.63617475241101087708365634860, −1.47016836472531453831984947977, −1.25671956635564698268715408649, 0, 0,
1.25671956635564698268715408649, 1.47016836472531453831984947977, 2.63617475241101087708365634860, 2.93187929445904833014945174614, 3.79029730786822254235881330119, 3.98060992217025302004451488127, 4.45508437457356945914192701442, 4.54388982970115817675053300559, 5.32470870015525549358797508304, 5.48100953375930169935458358780, 6.34517138040762474760387129009, 6.40017318611099806179872290760, 6.96984238991792902771881241245, 7.10276400475580769464366541150, 7.70170300688637950129612773536, 7.72806953602727379737167512769, 8.633896624027348567285064966310, 8.666309315043024180010044936232
