| L(s) = 1 | + 4·37-s − 2·49-s − 4·61-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 4·37-s − 2·49-s − 4·61-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8024183137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8024183137\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_2^2$ | \( 1 + T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 + T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−10.97108930542091516069013163656, −10.90046758494949597621111055440, −10.32128600765196702831252314017, −9.608954118362444963440514521674, −9.515808765871808328721034593563, −9.179892747410123030637517918515, −8.265626120990982086581753555569, −8.227481966722747312118115678022, −7.49978050550942334537889660406, −7.37828111450386903345772330147, −6.47695123701587543865650849036, −6.14228159995456275540848262986, −5.87962610494479260809048372873, −5.04945377720894180861970613440, −4.46416249146113780149850146048, −4.30905195764882491395254245485, −3.28880279911114015686970371616, −2.92870639926600146847447565313, −2.13681026124572579666661934196, −1.22422125775743083020646285934,
1.22422125775743083020646285934, 2.13681026124572579666661934196, 2.92870639926600146847447565313, 3.28880279911114015686970371616, 4.30905195764882491395254245485, 4.46416249146113780149850146048, 5.04945377720894180861970613440, 5.87962610494479260809048372873, 6.14228159995456275540848262986, 6.47695123701587543865650849036, 7.37828111450386903345772330147, 7.49978050550942334537889660406, 8.227481966722747312118115678022, 8.265626120990982086581753555569, 9.179892747410123030637517918515, 9.515808765871808328721034593563, 9.608954118362444963440514521674, 10.32128600765196702831252314017, 10.90046758494949597621111055440, 10.97108930542091516069013163656
