| L(s) = 1 | + 8·5-s + 6·7-s − 10·11-s − 10·13-s + 16·17-s + 6·19-s + 6·23-s + 44·25-s + 6·29-s + 8·31-s + 48·35-s + 12·37-s + 16·43-s − 2·47-s + 8·49-s + 16·53-s − 80·55-s + 6·59-s − 12·61-s − 80·65-s − 16·67-s − 60·77-s + 24·79-s − 2·83-s + 128·85-s − 60·91-s + 48·95-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 2.26·7-s − 3.01·11-s − 2.77·13-s + 3.88·17-s + 1.37·19-s + 1.25·23-s + 44/5·25-s + 1.11·29-s + 1.43·31-s + 8.11·35-s + 1.97·37-s + 2.43·43-s − 0.291·47-s + 8/7·49-s + 2.19·53-s − 10.7·55-s + 0.781·59-s − 1.53·61-s − 9.92·65-s − 1.95·67-s − 6.83·77-s + 2.70·79-s − 0.219·83-s + 13.8·85-s − 6.28·91-s + 4.92·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(20.12118023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.12118023\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T + 4 p T^{2} + 4 T^{3} - 89 T^{4} + 4 p T^{5} + 4 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 82 T^{3} + 136 T^{4} + 82 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 132 T^{3} + 959 T^{4} - 132 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 240 p T^{5} + 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 457 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T - 2 T^{2} - 32 T^{3} + 1411 T^{4} - 32 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 588 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 16 T + 65 T^{2} + 624 T^{3} - 8092 T^{4} + 624 p T^{5} + 65 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 148 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 976 T^{3} + 7378 T^{4} - 976 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T + 45 T^{2} + 594 T^{3} - 3376 T^{4} + 594 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 180 T^{2} + 1596 T^{3} + 15143 T^{4} + 1596 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 113 T^{2} + 384 T^{3} - 172 T^{4} + 384 p T^{5} + 113 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 328 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 109 T^{2} + 20 p T^{3} + 376 p T^{4} + 20 p^{2} T^{5} + 109 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.74392525762577163173916360405, −6.26058785291757868369048916566, −5.94502945978198527284105974404, −5.82278684955675445753518697344, −5.79299651406834563412061220869, −5.45051269111685758408101942678, −5.29149361343509990926232856118, −5.26038041375539002259844229219, −5.01393902384560110818772415562, −4.82157672420093057062033207925, −4.63684809192438005860293216043, −4.49574394085364304428198265458, −4.45945386841373724524389635188, −3.26830725285268289836609606098, −3.23911703973981658154019301280, −3.13671282698050593334454259811, −2.95694359390095642655836284027, −2.39243781198595182523525984267, −2.34044043373154937088283611177, −2.31410539464191925640222046093, −2.15028089707263274171209692966, −1.44186056819739029083951029326, −0.981273234775047120672720216413, −0.974007487622345031718137589208, −0.972848003299635026864511735236,
0.972848003299635026864511735236, 0.974007487622345031718137589208, 0.981273234775047120672720216413, 1.44186056819739029083951029326, 2.15028089707263274171209692966, 2.31410539464191925640222046093, 2.34044043373154937088283611177, 2.39243781198595182523525984267, 2.95694359390095642655836284027, 3.13671282698050593334454259811, 3.23911703973981658154019301280, 3.26830725285268289836609606098, 4.45945386841373724524389635188, 4.49574394085364304428198265458, 4.63684809192438005860293216043, 4.82157672420093057062033207925, 5.01393902384560110818772415562, 5.26038041375539002259844229219, 5.29149361343509990926232856118, 5.45051269111685758408101942678, 5.79299651406834563412061220869, 5.82278684955675445753518697344, 5.94502945978198527284105974404, 6.26058785291757868369048916566, 6.74392525762577163173916360405
