| L(s) = 1 | + 2·3-s + 2·5-s − 2·9-s + 8·13-s + 4·15-s + 4·17-s + 4·19-s + 4·23-s − 8·25-s − 6·27-s + 8·29-s − 10·31-s + 2·37-s + 16·39-s + 12·41-s − 4·43-s − 4·45-s − 14·47-s + 8·51-s − 10·53-s + 8·57-s + 6·59-s + 6·61-s + 16·65-s − 22·67-s + 8·69-s − 24·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 2/3·9-s + 2.21·13-s + 1.03·15-s + 0.970·17-s + 0.917·19-s + 0.834·23-s − 8/5·25-s − 1.15·27-s + 1.48·29-s − 1.79·31-s + 0.328·37-s + 2.56·39-s + 1.87·41-s − 0.609·43-s − 0.596·45-s − 2.04·47-s + 1.12·51-s − 1.37·53-s + 1.05·57-s + 0.781·59-s + 0.768·61-s + 1.98·65-s − 2.68·67-s + 0.963·69-s − 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.99225667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.99225667\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 2 T + 2 p T^{2} - 10 T^{3} + p^{3} T^{4} - 10 p T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 2 T + 12 T^{2} - 28 T^{3} + 73 T^{4} - 28 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 6 T^{2} - 38 T^{3} + 87 T^{4} - 38 p T^{5} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 8 T + 60 T^{2} - 252 T^{3} + 1106 T^{4} - 252 p T^{5} + 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 4 T + 26 T^{2} - 36 T^{3} + 403 T^{4} - 36 p T^{5} + 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 4 T + 54 T^{2} - 102 T^{3} + 1179 T^{4} - 102 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 4 T + 52 T^{2} - 126 T^{3} + 1209 T^{4} - 126 p T^{5} + 52 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 8 T + 96 T^{2} - 628 T^{3} + 3898 T^{4} - 628 p T^{5} + 96 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 10 T + 156 T^{2} + 966 T^{3} + 7761 T^{4} + 966 p T^{5} + 156 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 2 T + 56 T^{2} - 452 T^{3} + 1657 T^{4} - 452 p T^{5} + 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 12 T + 172 T^{2} - 1280 T^{3} + 10698 T^{4} - 1280 p T^{5} + 172 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 4 T + 140 T^{2} + 388 T^{3} + 8278 T^{4} + 388 p T^{5} + 140 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 14 T + 200 T^{2} + 1834 T^{3} + 14181 T^{4} + 1834 p T^{5} + 200 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 10 T + 56 T^{2} + 172 T^{3} - 287 T^{4} + 172 p T^{5} + 56 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 6 T + 206 T^{2} - 1018 T^{3} + 17443 T^{4} - 1018 p T^{5} + 206 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 6 T + 72 T^{2} + 548 T^{3} - 2215 T^{4} + 548 p T^{5} + 72 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 22 T + 390 T^{2} + 4482 T^{3} + 43271 T^{4} + 4482 p T^{5} + 390 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 16 T + 246 T^{2} - 1896 T^{3} + 19695 T^{4} - 1896 p T^{5} + 246 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 8 T + 216 T^{2} + 1418 T^{3} + 24813 T^{4} + 1418 p T^{5} + 216 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 16 T + 180 T^{2} - 368 T^{3} + 1462 T^{4} - 368 p T^{5} + 180 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 8 T + 294 T^{2} - 1480 T^{3} + 35263 T^{4} - 1480 p T^{5} + 294 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 16 T + 272 T^{2} - 2180 T^{3} + 27122 T^{4} - 2180 p T^{5} + 272 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.67312421232861167702898906473, −5.55985556844631211167867979657, −5.32281359774005942307262805723, −5.13221340947301227766004357514, −4.99360646064489514657976179925, −4.55782070893166410439417677334, −4.53245739816895385715987280226, −4.42476951921961133365251593850, −3.98645282215191927428159190887, −3.75352336424156833801393137667, −3.65913222738934344749096711143, −3.48044706781842251489616866112, −3.20471429709614522677564493489, −3.16456924687758569377770749395, −2.84301604656052056630297291108, −2.73602797829177014889888551735, −2.68779888046155920279400048010, −1.98956229125192414484384824594, −1.95836104495281545752392717787, −1.77106722111432211088540702459, −1.65332903842147545066988397216, −1.15745469359473766097538834575, −1.06571184746149593522241160896, −0.61170829864928691103398380328, −0.37132688251583681167626987070,
0.37132688251583681167626987070, 0.61170829864928691103398380328, 1.06571184746149593522241160896, 1.15745469359473766097538834575, 1.65332903842147545066988397216, 1.77106722111432211088540702459, 1.95836104495281545752392717787, 1.98956229125192414484384824594, 2.68779888046155920279400048010, 2.73602797829177014889888551735, 2.84301604656052056630297291108, 3.16456924687758569377770749395, 3.20471429709614522677564493489, 3.48044706781842251489616866112, 3.65913222738934344749096711143, 3.75352336424156833801393137667, 3.98645282215191927428159190887, 4.42476951921961133365251593850, 4.53245739816895385715987280226, 4.55782070893166410439417677334, 4.99360646064489514657976179925, 5.13221340947301227766004357514, 5.32281359774005942307262805723, 5.55985556844631211167867979657, 5.67312421232861167702898906473
