| L(s) = 1 | − 4·5-s − 4·7-s + 2·9-s + 4·13-s + 4·17-s + 12·23-s + 2·25-s − 8·29-s + 16·35-s − 12·37-s + 4·43-s − 8·45-s + 20·47-s + 8·49-s − 12·53-s − 16·59-s + 24·61-s − 8·63-s − 16·65-s + 12·67-s + 4·73-s − 5·81-s − 4·83-s − 16·85-s + 40·89-s − 16·91-s + 4·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.51·7-s + 2/3·9-s + 1.10·13-s + 0.970·17-s + 2.50·23-s + 2/5·25-s − 1.48·29-s + 2.70·35-s − 1.97·37-s + 0.609·43-s − 1.19·45-s + 2.91·47-s + 8/7·49-s − 1.64·53-s − 2.08·59-s + 3.07·61-s − 1.00·63-s − 1.98·65-s + 1.46·67-s + 0.468·73-s − 5/9·81-s − 0.439·83-s − 1.73·85-s + 4.23·89-s − 1.67·91-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6857859563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6857859563\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 20 T^{3} + 46 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 4 T^{3} - 194 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 178 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2542 T^{4} - 444 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 468 T^{3} + 3038 T^{4} + 468 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 164 T^{3} + 3358 T^{4} - 164 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1860 T^{3} + 15182 T^{4} - 1860 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} + 180 T^{3} - 6274 T^{4} + 180 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 13286 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 44 T^{3} - 3602 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 196 T^{3} + 3646 T^{4} + 196 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.806461743169845918341700428043, −9.732839906929769034962417763783, −9.346263315336985112520125717747, −9.130846668355472812328543584266, −8.748565097446662546464856425687, −8.695653315685768804284715080715, −8.200516864596251416301285236798, −7.82123796283941717276572096540, −7.60887851409104569524662869298, −7.31485467661676591988882119203, −7.18618872469779098628965231316, −6.74642893262408834765087481994, −6.46407157083440294202153517046, −6.23665494021807760978154229527, −5.63810030527158129573672484423, −5.25571651160581903451386098816, −5.25311838401986642963567547365, −4.53639411472953260540295672291, −3.89332655317309448156251699698, −3.83608124445814178280715476541, −3.72268575311739549786412265139, −3.08490949477847791697644051963, −2.85965166053531634219261317474, −1.84911852831014022172884294492, −0.830984185157060022921602174057,
0.830984185157060022921602174057, 1.84911852831014022172884294492, 2.85965166053531634219261317474, 3.08490949477847791697644051963, 3.72268575311739549786412265139, 3.83608124445814178280715476541, 3.89332655317309448156251699698, 4.53639411472953260540295672291, 5.25311838401986642963567547365, 5.25571651160581903451386098816, 5.63810030527158129573672484423, 6.23665494021807760978154229527, 6.46407157083440294202153517046, 6.74642893262408834765087481994, 7.18618872469779098628965231316, 7.31485467661676591988882119203, 7.60887851409104569524662869298, 7.82123796283941717276572096540, 8.200516864596251416301285236798, 8.695653315685768804284715080715, 8.748565097446662546464856425687, 9.130846668355472812328543584266, 9.346263315336985112520125717747, 9.732839906929769034962417763783, 9.806461743169845918341700428043
