| L(s) = 1 | + 4·4-s + 4·7-s + 4·16-s − 14·25-s + 16·28-s + 20·37-s + 20·43-s − 2·49-s − 16·64-s + 8·67-s − 52·79-s − 56·100-s − 28·109-s + 16·112-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 80·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 80·172-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.51·7-s + 16-s − 2.79·25-s + 3.02·28-s + 3.28·37-s + 3.04·43-s − 2/7·49-s − 2·64-s + 0.977·67-s − 5.85·79-s − 5.59·100-s − 2.68·109-s + 1.51·112-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.57·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 6.09·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.863349295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.863349295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 163 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.341091374372791732670732620300, −8.909766763618312675979689306802, −8.507092058131594793711533661069, −8.362479658156968219551841550469, −8.006987996081415846114091344901, −7.72969249655452372306215068052, −7.57985669344396046469323545879, −7.42781759225687197034043682863, −7.07752398678913708314848478297, −6.75682140315435819717536517767, −6.49192134645838242014543461484, −6.02741144301485468951493417722, −5.81826367315714878428283556666, −5.62524808166462633392988730764, −5.60233471199409282489903453001, −4.63727216477519667024161427335, −4.47567587387250422591413427531, −4.27294115020486697734705743889, −4.06322499893154788370364151799, −3.26072537580287945895711806105, −2.87821755845660599453994694347, −2.50285706865570168890423261532, −2.11618973299712712027780402745, −1.83112991167314617853490303370, −1.21390625503354902557172281617,
1.21390625503354902557172281617, 1.83112991167314617853490303370, 2.11618973299712712027780402745, 2.50285706865570168890423261532, 2.87821755845660599453994694347, 3.26072537580287945895711806105, 4.06322499893154788370364151799, 4.27294115020486697734705743889, 4.47567587387250422591413427531, 4.63727216477519667024161427335, 5.60233471199409282489903453001, 5.62524808166462633392988730764, 5.81826367315714878428283556666, 6.02741144301485468951493417722, 6.49192134645838242014543461484, 6.75682140315435819717536517767, 7.07752398678913708314848478297, 7.42781759225687197034043682863, 7.57985669344396046469323545879, 7.72969249655452372306215068052, 8.006987996081415846114091344901, 8.362479658156968219551841550469, 8.507092058131594793711533661069, 8.909766763618312675979689306802, 9.341091374372791732670732620300
