| L(s) = 1 | − 6·9-s − 24·19-s − 2·25-s − 4·29-s − 24·31-s + 24·37-s − 2·49-s − 8·53-s + 9·81-s − 48·83-s − 48·103-s − 36·109-s + 8·113-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 144·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2·9-s − 5.50·19-s − 2/5·25-s − 0.742·29-s − 4.31·31-s + 3.94·37-s − 2/7·49-s − 1.09·53-s + 81-s − 5.26·83-s − 4.72·103-s − 3.44·109-s + 0.752·113-s + 2.72·121-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 + 0.769·169-s + 11.0·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09763122555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09763122555\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 315 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11190 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7590 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 94 T^{2} + 3891 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 154 T^{2} + 20859 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.25233915471763944964639059085, −6.19632323610922851707554678945, −5.98520033957741246974910471758, −5.67525194776271474614683888872, −5.66588584184211456407633101647, −5.66058028228676386185515788969, −5.34292051295322053213082154634, −4.75636472115034884278332276984, −4.71838031222893140900605262488, −4.44039300845477006303835967867, −4.21994340056006105512705195320, −4.18478198315412092916500842794, −3.73124216282188634410428064241, −3.66677072509821674448823513781, −3.59314970336008862572585348162, −2.87098738860332728352054479488, −2.67357958875964586914011678103, −2.56037198850972309905948826480, −2.55993227511688063479947578651, −2.05101466215346182066550663208, −1.77202516625312643078175045202, −1.61391233479248321389818645081, −1.23549733927614951666535601286, −0.22766335112949957398417250885, −0.15518732293845840692107735632,
0.15518732293845840692107735632, 0.22766335112949957398417250885, 1.23549733927614951666535601286, 1.61391233479248321389818645081, 1.77202516625312643078175045202, 2.05101466215346182066550663208, 2.55993227511688063479947578651, 2.56037198850972309905948826480, 2.67357958875964586914011678103, 2.87098738860332728352054479488, 3.59314970336008862572585348162, 3.66677072509821674448823513781, 3.73124216282188634410428064241, 4.18478198315412092916500842794, 4.21994340056006105512705195320, 4.44039300845477006303835967867, 4.71838031222893140900605262488, 4.75636472115034884278332276984, 5.34292051295322053213082154634, 5.66058028228676386185515788969, 5.66588584184211456407633101647, 5.67525194776271474614683888872, 5.98520033957741246974910471758, 6.19632323610922851707554678945, 6.25233915471763944964639059085
