| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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
−5.95646203867408596420462089869, −5.66373512038698636977936137286, −5.62897212247227784101093436836, −5.40874795183790324590013976173, −5.34980699769256703276392089990, −4.73545005867717265065072720047, −4.64019516583313422808916042586, −4.47141800036798768317385385863, −4.33013755401181163567395385472, −3.95122225716364527548652146819, −3.79291408372660069324239826330, −3.59686935022065176530043590751, −3.46313004564590165931277255020, −3.19284051071542070693162123401, −3.03434879977726604559962535629, −2.97376422352147085524106343460, −2.94977437651244492772893670245, −2.35008290452733810692599701387, −2.13304438889802667419608646203, −2.11410105784859999534708810100, −2.05301745098409672903058895852, −1.53664001161362147725470177768, −1.25639945374334753573577170555, −0.938450096941583789094546843435, −0.811222802022697145909076791419, 0, 0, 0, 0,
0.811222802022697145909076791419, 0.938450096941583789094546843435, 1.25639945374334753573577170555, 1.53664001161362147725470177768, 2.05301745098409672903058895852, 2.11410105784859999534708810100, 2.13304438889802667419608646203, 2.35008290452733810692599701387, 2.94977437651244492772893670245, 2.97376422352147085524106343460, 3.03434879977726604559962535629, 3.19284051071542070693162123401, 3.46313004564590165931277255020, 3.59686935022065176530043590751, 3.79291408372660069324239826330, 3.95122225716364527548652146819, 4.33013755401181163567395385472, 4.47141800036798768317385385863, 4.64019516583313422808916042586, 4.73545005867717265065072720047, 5.34980699769256703276392089990, 5.40874795183790324590013976173, 5.62897212247227784101093436836, 5.66373512038698636977936137286, 5.95646203867408596420462089869
