| L(s) = 1 | + 4·2-s + 2·3-s + 10·4-s − 4·5-s + 8·6-s + 4·7-s + 20·8-s − 9-s − 16·10-s + 20·12-s + 13-s + 16·14-s − 8·15-s + 35·16-s + 2·17-s − 4·18-s − 3·19-s − 40·20-s + 8·21-s − 8·23-s + 40·24-s + 10·25-s + 4·26-s − 2·27-s + 40·28-s + 15·29-s − 32·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 5·4-s − 1.78·5-s + 3.26·6-s + 1.51·7-s + 7.07·8-s − 1/3·9-s − 5.05·10-s + 5.77·12-s + 0.277·13-s + 4.27·14-s − 2.06·15-s + 35/4·16-s + 0.485·17-s − 0.942·18-s − 0.688·19-s − 8.94·20-s + 1.74·21-s − 1.66·23-s + 8.16·24-s + 2·25-s + 0.784·26-s − 0.384·27-s + 7.55·28-s + 2.78·29-s − 5.84·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(122.5242991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(122.5242991\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 23 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 3 p T^{2} - 23 T^{3} + 672 T^{4} - 23 p T^{5} + 3 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 3 p T^{2} - 64 T^{3} + 1137 T^{4} - 64 p T^{5} + 3 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 35 T^{2} + 153 T^{3} + 844 T^{4} + 153 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 177 T^{2} - 1395 T^{3} + 8628 T^{4} - 1395 p T^{5} + 177 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 88 T^{2} - 46 T^{3} + 3838 T^{4} - 46 p T^{5} + 88 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 129 T^{2} - 1001 T^{3} + 8240 T^{4} - 1001 p T^{5} + 129 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 T + 105 T^{2} - 605 T^{3} + 6408 T^{4} - 605 p T^{5} + 105 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T - 13 T^{2} - 195 T^{3} + 4964 T^{4} - 195 p T^{5} - 13 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 168 T^{2} - 30 p T^{3} + 12254 T^{4} - 30 p^{2} T^{5} + 168 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 13 T + 151 T^{2} - 1427 T^{3} + 13660 T^{4} - 1427 p T^{5} + 151 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 26 T + 444 T^{2} - 5086 T^{3} + 750 p T^{4} - 5086 p T^{5} + 444 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 13 T + 287 T^{2} + 2415 T^{3} + 29196 T^{4} + 2415 p T^{5} + 287 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 13 T + 233 T^{2} + 1991 T^{3} + 21640 T^{4} + 1991 p T^{5} + 233 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 18 T + 267 T^{2} + 3288 T^{3} + 29193 T^{4} + 3288 p T^{5} + 267 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 387 T^{2} - 3675 T^{3} + 48708 T^{4} - 3675 p T^{5} + 387 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 217 T^{2} + 742 T^{3} + 22323 T^{4} + 742 p T^{5} + 217 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 7 T + 251 T^{2} + 1223 T^{3} + 29920 T^{4} + 1223 p T^{5} + 251 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 5 T + 49 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.49909124357501784793813115330, −5.13885553149263524514628524512, −5.03609861527455050110544065850, −4.66764473060443797495167573215, −4.62283714087448810873913752123, −4.48659605462219639862144507841, −4.31591676873505041884075150265, −4.21619281443108146878421706784, −4.14536815953383592872552742619, −3.75500977619252813889459673608, −3.53279989610734530522788604446, −3.51314921031663299894449343944, −3.44804057570829548576317031608, −2.85908030240174506612132084873, −2.78574945927173843153456022270, −2.70621969050571537283438950633, −2.67346039142872270468538284733, −2.18494701443797804958311475323, −2.17105705241084043025777071337, −1.87630047434555690430786727106, −1.67766863353070213958368020773, −1.12206128739584985024541740578, −0.933933928100366358608002462078, −0.64768992505244579987534307702, −0.61977358199306326016596966969,
0.61977358199306326016596966969, 0.64768992505244579987534307702, 0.933933928100366358608002462078, 1.12206128739584985024541740578, 1.67766863353070213958368020773, 1.87630047434555690430786727106, 2.17105705241084043025777071337, 2.18494701443797804958311475323, 2.67346039142872270468538284733, 2.70621969050571537283438950633, 2.78574945927173843153456022270, 2.85908030240174506612132084873, 3.44804057570829548576317031608, 3.51314921031663299894449343944, 3.53279989610734530522788604446, 3.75500977619252813889459673608, 4.14536815953383592872552742619, 4.21619281443108146878421706784, 4.31591676873505041884075150265, 4.48659605462219639862144507841, 4.62283714087448810873913752123, 4.66764473060443797495167573215, 5.03609861527455050110544065850, 5.13885553149263524514628524512, 5.49909124357501784793813115330
