| L(s) = 1 | + 8·7-s − 16·11-s − 16·17-s + 16·43-s + 16·49-s + 16·53-s + 32·59-s + 40·61-s + 32·67-s − 128·77-s − 8·103-s + 40·109-s + 16·113-s − 128·119-s + 120·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 4.82·11-s − 3.88·17-s + 2.43·43-s + 16/7·49-s + 2.19·53-s + 4.16·59-s + 5.12·61-s + 3.90·67-s − 14.5·77-s − 0.788·103-s + 3.83·109-s + 1.50·113-s − 11.7·119-s + 10.9·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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 + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.003005601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.003005601\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 706 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_4\times C_2$ | \( 1 - 68 T^{2} + 2086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 + 28 T^{2} + 966 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 2338 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 418 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 16 T + 180 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 14406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4614 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 9366 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 15846 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.40229658067524275772792384544, −5.28808586165357160690146539224, −5.14477394395156324200920275953, −5.13480640253248385313256220144, −5.01461706523697598330738113264, −4.47503098014702205048013770220, −4.47349032630588005411398639298, −4.38161078184177786715753935386, −4.33826924227061645563548630276, −3.97908113189217035425455021720, −3.61867248576344371556626724078, −3.48020753664247960777942961425, −3.37383893043293943385982784151, −2.57826000951774508594017075066, −2.53912892819140806009814650276, −2.53911773803423443614873540490, −2.53490894016201736268093226905, −2.11156603522544131714565743801, −2.04325776151440029301829325033, −1.96752291682562092640747186340, −1.68160808649729053898211840289, −0.973973985560210054859624720364, −0.60578696835215163020183016291, −0.57022588747924131615182560577, −0.48950863938331367916346157366,
0.48950863938331367916346157366, 0.57022588747924131615182560577, 0.60578696835215163020183016291, 0.973973985560210054859624720364, 1.68160808649729053898211840289, 1.96752291682562092640747186340, 2.04325776151440029301829325033, 2.11156603522544131714565743801, 2.53490894016201736268093226905, 2.53911773803423443614873540490, 2.53912892819140806009814650276, 2.57826000951774508594017075066, 3.37383893043293943385982784151, 3.48020753664247960777942961425, 3.61867248576344371556626724078, 3.97908113189217035425455021720, 4.33826924227061645563548630276, 4.38161078184177786715753935386, 4.47349032630588005411398639298, 4.47503098014702205048013770220, 5.01461706523697598330738113264, 5.13480640253248385313256220144, 5.14477394395156324200920275953, 5.28808586165357160690146539224, 5.40229658067524275772792384544
