| 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 − 4.79·15-s − 0.421·19-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(34.27520538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(34.27520538\) |
| \(L(2)\) |
|
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
−7.67448195319248945577004048149, −7.34482980624141117809351449002, −7.29456403412697544876071671020, −6.88813167569989772737104925355, −6.70494959450269473508175319731, −6.45222081320861973693901882149, −6.26884864496769884922748216417, −6.23273940440639986766501663639, −5.49989184037788732915779209031, −5.05398933174623130114866825982, −4.69977025696506042830821918527, −4.56866169049821793774190876385, −4.26454530284287362355828277931, −3.85027344442195721233138870437, −3.79448366997570974918677648936, −3.70131455162801110440201748004, −3.55177177308958986228403185427, −3.02964242668641056997276204255, −2.60027015351911080305266003717, −2.56718656808557055455405157069, −2.49581172823459236844651768699, −1.71051482389479678916026393753, −1.37819337430813906522954436498, −0.980422110614424418984232618323, −0.837743817370796911723038823222,
0.837743817370796911723038823222, 0.980422110614424418984232618323, 1.37819337430813906522954436498, 1.71051482389479678916026393753, 2.49581172823459236844651768699, 2.56718656808557055455405157069, 2.60027015351911080305266003717, 3.02964242668641056997276204255, 3.55177177308958986228403185427, 3.70131455162801110440201748004, 3.79448366997570974918677648936, 3.85027344442195721233138870437, 4.26454530284287362355828277931, 4.56866169049821793774190876385, 4.69977025696506042830821918527, 5.05398933174623130114866825982, 5.49989184037788732915779209031, 6.23273940440639986766501663639, 6.26884864496769884922748216417, 6.45222081320861973693901882149, 6.70494959450269473508175319731, 6.88813167569989772737104925355, 7.29456403412697544876071671020, 7.34482980624141117809351449002, 7.67448195319248945577004048149
