| L(s) = 1 | − 4·3-s − 4-s + 4·5-s + 10·9-s − 4·11-s + 4·12-s − 8·13-s − 16·15-s + 3·16-s + 2·17-s − 10·19-s − 4·20-s − 2·23-s + 10·25-s − 20·27-s − 2·29-s − 24·31-s + 16·33-s − 10·36-s + 8·37-s + 32·39-s + 6·43-s + 4·44-s + 40·45-s − 4·47-s − 12·48-s − 8·51-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1/2·4-s + 1.78·5-s + 10/3·9-s − 1.20·11-s + 1.15·12-s − 2.21·13-s − 4.13·15-s + 3/4·16-s + 0.485·17-s − 2.29·19-s − 0.894·20-s − 0.417·23-s + 2·25-s − 3.84·27-s − 0.371·29-s − 4.31·31-s + 2.78·33-s − 5/3·36-s + 1.31·37-s + 5.12·39-s + 0.914·43-s + 0.603·44-s + 5.96·45-s − 0.583·47-s − 1.73·48-s − 1.12·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + T^{2} - p T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 50 T^{2} + 240 T^{3} + 1002 T^{4} + 240 p T^{5} + 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10 T + 37 T^{2} - 78 T^{3} - 916 T^{4} - 78 p T^{5} + 37 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 21 T^{2} - 98 T^{3} - 108 T^{4} - 98 p T^{5} + 21 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 71 T^{2} + 210 T^{3} + 2432 T^{4} + 210 p T^{5} + 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 326 T^{2} + 2928 T^{3} + 19090 T^{4} + 2928 p T^{5} + 326 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 114 T^{2} - 688 T^{3} + 6282 T^{4} - 688 p T^{5} + 114 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 38 T^{2} + 402 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 107 T^{2} - 470 T^{3} + 5520 T^{4} - 470 p T^{5} + 107 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 54 T^{2} + 452 T^{3} + 786 T^{4} + 452 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 14 T + 161 T^{2} - 1198 T^{3} + 10012 T^{4} - 1198 p T^{5} + 161 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 203 T^{2} - 238 T^{3} + 16912 T^{4} - 238 p T^{5} + 203 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 121 T^{2} + 866 T^{3} + 9380 T^{4} + 866 p T^{5} + 121 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 176 T^{2} + 1176 T^{3} + 17358 T^{4} + 1176 p T^{5} + 176 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 258 T^{2} + 26682 T^{4} + 258 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 112 T^{2} - 148 T^{3} + 4318 T^{4} - 148 p T^{5} + 112 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 290 T^{2} + 33466 T^{4} + 290 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 117 T^{2} + 1478 T^{3} + 148 p T^{4} + 1478 p T^{5} + 117 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 18 T + 255 T^{2} + 2594 T^{3} + 29096 T^{4} + 2594 p T^{5} + 255 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 111 T^{2} + 834 T^{3} - 984 T^{4} + 834 p T^{5} + 111 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.83207795858468783915005197241, −5.78339606392488532577430967953, −5.47115568188599162391086350835, −5.38244483100691000387115920840, −5.37231145888572410734528325514, −4.86959263398688257448719798465, −4.77908725132035431221204157094, −4.77626334312120929573893826885, −4.56837043746028316255212738169, −4.24008438646064645870117710661, −4.11346249417661852671072800638, −4.04876268369709162849911358678, −3.68499352782593120023455397383, −3.35800256530663087693135667142, −3.17216302813275361957443111902, −3.05886168668764964740543550350, −2.69461193718203973157661443834, −2.30445041675029733728677209302, −2.14714306434905729750072744198, −2.07277635868679862834076253162, −2.00757499957312153912487989695, −1.80485860913552044095952151107, −1.09993917645760893655179320298, −1.08180676284564603857244051364, −1.01170515727297902441477778509, 0, 0, 0, 0,
1.01170515727297902441477778509, 1.08180676284564603857244051364, 1.09993917645760893655179320298, 1.80485860913552044095952151107, 2.00757499957312153912487989695, 2.07277635868679862834076253162, 2.14714306434905729750072744198, 2.30445041675029733728677209302, 2.69461193718203973157661443834, 3.05886168668764964740543550350, 3.17216302813275361957443111902, 3.35800256530663087693135667142, 3.68499352782593120023455397383, 4.04876268369709162849911358678, 4.11346249417661852671072800638, 4.24008438646064645870117710661, 4.56837043746028316255212738169, 4.77626334312120929573893826885, 4.77908725132035431221204157094, 4.86959263398688257448719798465, 5.37231145888572410734528325514, 5.38244483100691000387115920840, 5.47115568188599162391086350835, 5.78339606392488532577430967953, 5.83207795858468783915005197241
