| L(s) = 1 | − 12·3-s − 6·5-s + 6·7-s + 90·9-s − 18·11-s + 14·13-s + 72·15-s + 8·19-s − 72·21-s − 30·23-s + 3·25-s − 540·27-s − 6·29-s − 74·31-s + 216·33-s − 36·35-s + 120·37-s − 168·39-s − 138·41-s + 10·43-s − 540·45-s − 174·47-s + 11·49-s + 108·55-s − 96·57-s − 18·59-s + 62·61-s + ⋯ |
| L(s) = 1 | − 4·3-s − 6/5·5-s + 6/7·7-s + 10·9-s − 1.63·11-s + 1.07·13-s + 24/5·15-s + 8/19·19-s − 3.42·21-s − 1.30·23-s + 3/25·25-s − 20·27-s − 0.206·29-s − 2.38·31-s + 6.54·33-s − 1.02·35-s + 3.24·37-s − 4.30·39-s − 3.36·41-s + 0.232·43-s − 12·45-s − 3.70·47-s + 0.224·49-s + 1.96·55-s − 1.68·57-s − 0.305·59-s + 1.01·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07336311796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07336311796\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 6 T + 33 T^{2} + 126 T^{3} + 116 T^{4} + 126 p^{2} T^{5} + 33 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 25 T^{2} + 522 T^{3} - 4044 T^{4} + 522 p^{2} T^{5} + 25 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 18 T + 249 T^{2} + 2538 T^{3} + 18308 T^{4} + 2538 p^{2} T^{5} + 249 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 14 T - 95 T^{2} + 658 T^{3} + 22996 T^{4} + 658 p^{2} T^{5} - 95 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 516 T^{2} + 135302 T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 18 p T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 30 T + 1401 T^{2} + 33030 T^{3} + 1091060 T^{4} + 33030 p^{2} T^{5} + 1401 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + 1409 T^{2} + 8382 T^{3} + 1254420 T^{4} + 8382 p^{2} T^{5} + 1409 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 74 T + 2281 T^{2} + 94202 T^{3} + 4022068 T^{4} + 94202 p^{2} T^{5} + 2281 p^{4} T^{6} + 74 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 60 T + 3254 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 69 T + 3268 T^{2} + 69 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 10 T - 2087 T^{2} + 15110 T^{3} + 1179268 T^{4} + 15110 p^{2} T^{5} - 2087 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 174 T + 16745 T^{2} + 1157622 T^{3} + 61675956 T^{4} + 1157622 p^{2} T^{5} + 16745 p^{4} T^{6} + 174 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 996 T^{2} - 9136858 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 6969 T^{2} + 123498 T^{3} + 35331908 T^{4} + 123498 p^{2} T^{5} + 6969 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 62 T - 4463 T^{2} - 53630 T^{3} + 40856884 T^{4} - 53630 p^{2} T^{5} - 4463 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 22 T + 985 T^{2} - 208538 T^{3} - 22072796 T^{4} - 208538 p^{2} T^{5} + 985 p^{4} T^{6} + 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 16452 T^{2} + 117605702 T^{4} - 16452 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 20 T + 7302 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 86 T - 2231 T^{2} + 245530 T^{3} + 7570612 T^{4} + 245530 p^{2} T^{5} - 2231 p^{4} T^{6} - 86 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 66 T + 9321 T^{2} + 519354 T^{3} + 24465668 T^{4} + 519354 p^{2} T^{5} + 9321 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 25924 T^{2} + 285535302 T^{4} - 25924 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 242 T + 25489 T^{2} - 3450194 T^{3} + 454397668 T^{4} - 3450194 p^{2} T^{5} + 25489 p^{4} T^{6} - 242 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.34960226281422822109811851782, −7.25040704541379506212981675515, −7.18088318192835932446386344212, −6.50579514993457588049681475747, −6.41293590922224388091077974071, −6.34584305542696086593599559937, −6.02252634840992423638958316782, −5.91927850587679058184439227326, −5.33997725320508341351148742371, −5.28927594871518790996437788931, −5.17320620845654811577059928492, −4.95056599524645833715040198035, −4.74419241719274156209079354953, −4.27325879332092720008578462445, −4.23563988341693451252665424927, −3.78551219304738798781646494447, −3.57985217097449987865529813238, −3.44296817418871692739443809990, −2.69983270747904738958380692984, −1.94701430534579390970176225435, −1.73585402323493009765656508283, −1.62341324005338842498036189102, −0.971262284789838333069819944982, −0.50961658483301262789072033359, −0.13172050777226013689183905538,
0.13172050777226013689183905538, 0.50961658483301262789072033359, 0.971262284789838333069819944982, 1.62341324005338842498036189102, 1.73585402323493009765656508283, 1.94701430534579390970176225435, 2.69983270747904738958380692984, 3.44296817418871692739443809990, 3.57985217097449987865529813238, 3.78551219304738798781646494447, 4.23563988341693451252665424927, 4.27325879332092720008578462445, 4.74419241719274156209079354953, 4.95056599524645833715040198035, 5.17320620845654811577059928492, 5.28927594871518790996437788931, 5.33997725320508341351148742371, 5.91927850587679058184439227326, 6.02252634840992423638958316782, 6.34584305542696086593599559937, 6.41293590922224388091077974071, 6.50579514993457588049681475747, 7.18088318192835932446386344212, 7.25040704541379506212981675515, 7.34960226281422822109811851782
