| L(s) = 1 | − 4·2-s + 4·4-s + 4·5-s + 16·8-s − 6·9-s − 16·10-s + 20·13-s − 64·16-s − 80·17-s + 24·18-s + 16·20-s − 24·25-s − 80·26-s + 112·29-s + 64·32-s + 320·34-s − 24·36-s − 40·37-s + 64·40-s − 8·41-s − 24·45-s − 8·49-s + 96·50-s + 80·52-s + 52·53-s − 448·58-s − 4·61-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 4/5·5-s + 2·8-s − 2/3·9-s − 8/5·10-s + 1.53·13-s − 4·16-s − 4.70·17-s + 4/3·18-s + 4/5·20-s − 0.959·25-s − 3.07·26-s + 3.86·29-s + 2·32-s + 9.41·34-s − 2/3·36-s − 1.08·37-s + 8/5·40-s − 0.195·41-s − 0.533·45-s − 0.163·49-s + 1.91·50-s + 1.53·52-s + 0.981·53-s − 7.72·58-s − 0.0655·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4044959014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4044959014\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + 18 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 3630 T^{4} + 8 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 10 T + 330 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 40 T + 846 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 628 T^{2} + 340230 T^{4} - 628 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1192 T^{2} + 771150 T^{4} - 1192 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 56 T + 2334 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 651846 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 20 T + 2310 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T - 1386 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 3844 T^{2} + 9315174 T^{4} - 3844 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6568 T^{2} + 19677966 T^{4} - 6568 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 26 T + 5490 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6580 T^{2} + 33519174 T^{4} - 6580 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 4770 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 5090 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12616 T^{2} + 80339214 T^{4} - 12616 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 124 T + 9750 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 40 p T^{2} - 3030738 T^{4} - 40 p^{5} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 7970 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 292 T + 36630 T^{2} + 292 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 304 T + 40734 T^{2} - 304 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−9.601982183708599305463853459271, −9.025409014267321491101276828609, −8.784797082128610098993963790980, −8.755007666286781275135792556189, −8.494129318940361482779672913276, −8.431738694084789791469104744428, −8.285230064489809864411120973835, −7.53236642497542807058389878783, −7.36088472163341506724623016989, −6.96595809543905205644989023848, −6.55932500387201033714313396907, −6.48338041316652082212429707920, −6.22175265381183989063741286740, −5.87530289969919848971704589453, −5.20118511433423897695568674561, −4.77466957829398070274686097400, −4.66607232909690556237524073255, −4.38206677650804563255956093389, −3.88416871261637369566472642143, −3.45213666022171266527022024432, −2.57395257737984768566536002446, −2.13480215132067126613102776406, −2.00484079168505329363101560549, −1.09903719916065301487479286413, −0.40816399520615971761058231308,
0.40816399520615971761058231308, 1.09903719916065301487479286413, 2.00484079168505329363101560549, 2.13480215132067126613102776406, 2.57395257737984768566536002446, 3.45213666022171266527022024432, 3.88416871261637369566472642143, 4.38206677650804563255956093389, 4.66607232909690556237524073255, 4.77466957829398070274686097400, 5.20118511433423897695568674561, 5.87530289969919848971704589453, 6.22175265381183989063741286740, 6.48338041316652082212429707920, 6.55932500387201033714313396907, 6.96595809543905205644989023848, 7.36088472163341506724623016989, 7.53236642497542807058389878783, 8.285230064489809864411120973835, 8.431738694084789791469104744428, 8.494129318940361482779672913276, 8.755007666286781275135792556189, 8.784797082128610098993963790980, 9.025409014267321491101276828609, 9.601982183708599305463853459271
