| L(s) = 1 | − 2·5-s + 4·7-s − 2·11-s − 8·17-s − 4·19-s − 2·23-s − 4·25-s − 8·29-s + 4·31-s − 8·35-s − 4·37-s − 2·41-s − 8·43-s − 12·47-s + 10·49-s − 16·53-s + 4·55-s − 12·59-s − 8·61-s + 8·67-s + 18·71-s + 8·73-s − 8·77-s + 16·85-s − 18·89-s + 8·95-s + 8·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 0.603·11-s − 1.94·17-s − 0.917·19-s − 0.417·23-s − 4/5·25-s − 1.48·29-s + 0.718·31-s − 1.35·35-s − 0.657·37-s − 0.312·41-s − 1.21·43-s − 1.75·47-s + 10/7·49-s − 2.19·53-s + 0.539·55-s − 1.56·59-s − 1.02·61-s + 0.977·67-s + 2.13·71-s + 0.936·73-s − 0.911·77-s + 1.73·85-s − 1.90·89-s + 0.820·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 2 T + 8 T^{2} + 24 T^{3} + 33 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 20 T^{2} - 20 T^{3} + 125 T^{4} - 20 p T^{5} + 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 16 T^{2} - 32 T^{3} + 126 T^{4} - 32 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 8 T + 44 T^{2} + 184 T^{3} + 854 T^{4} + 184 p T^{5} + 44 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 4 T + 34 T^{2} + 72 T^{3} + 699 T^{4} + 72 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 8 T^{2} + 4 T^{3} + 761 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 8 T + 80 T^{2} + 296 T^{3} + 2206 T^{4} + 296 p T^{5} + 80 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 4 T + 70 T^{2} - 448 T^{3} + 2407 T^{4} - 448 p T^{5} + 70 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 4 T + 118 T^{2} + 344 T^{3} + 5975 T^{4} + 344 p T^{5} + 118 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 2 T + 92 T^{2} + 80 T^{3} + 4093 T^{4} + 80 p T^{5} + 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 8 T + 160 T^{2} + 952 T^{3} + 10046 T^{4} + 952 p T^{5} + 160 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 12 T + 200 T^{2} + 1484 T^{3} + 13950 T^{4} + 1484 p T^{5} + 200 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 12 T + 248 T^{2} + 1916 T^{3} + 21870 T^{4} + 1916 p T^{5} + 248 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 8 T + 100 T^{2} + 440 T^{3} + 3478 T^{4} + 440 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 8 T + 88 T^{2} - 8 T^{3} + 814 T^{4} - 8 p T^{5} + 88 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 18 T + 296 T^{2} - 2724 T^{3} + 27369 T^{4} - 2724 p T^{5} + 296 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 8 T + 136 T^{2} - 1032 T^{3} + 11166 T^{4} - 1032 p T^{5} + 136 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 160 T^{2} + 848 T^{3} + 11886 T^{4} + 848 p T^{5} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 40 T^{2} + 16 T^{3} + 13134 T^{4} + 16 p T^{5} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 18 T + 404 T^{2} + 4416 T^{3} + 54549 T^{4} + 4416 p T^{5} + 404 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 8 T + 244 T^{2} - 2200 T^{3} + 29798 T^{4} - 2200 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.14260017171135465786573044022, −5.84915774576260205627800573054, −5.46832726969435100595412486583, −5.45544449305825601215247146264, −5.23190982466024415953290872661, −5.07951553987441292718820215810, −4.89339692215446714803052102984, −4.85335956505728573845502537659, −4.43271897365286880668708882343, −4.37647022448244239499533932173, −4.14623802162842325790009360883, −3.96372480359134478396832746046, −3.89900381959242159029832469977, −3.50147236565789978357316949051, −3.35389309763696306449875618557, −3.33152625836231481214124990423, −2.73770170299340144041187481799, −2.53438157053937381774033938180, −2.48048763121543458375746276454, −2.22526209014727458417452950301, −2.05885920089271034156037459578, −1.51536851296709074778771312522, −1.49212911354658975128834941193, −1.43097230627553570402284655104, −1.05574691377499523292259938640, 0, 0, 0, 0,
1.05574691377499523292259938640, 1.43097230627553570402284655104, 1.49212911354658975128834941193, 1.51536851296709074778771312522, 2.05885920089271034156037459578, 2.22526209014727458417452950301, 2.48048763121543458375746276454, 2.53438157053937381774033938180, 2.73770170299340144041187481799, 3.33152625836231481214124990423, 3.35389309763696306449875618557, 3.50147236565789978357316949051, 3.89900381959242159029832469977, 3.96372480359134478396832746046, 4.14623802162842325790009360883, 4.37647022448244239499533932173, 4.43271897365286880668708882343, 4.85335956505728573845502537659, 4.89339692215446714803052102984, 5.07951553987441292718820215810, 5.23190982466024415953290872661, 5.45544449305825601215247146264, 5.46832726969435100595412486583, 5.84915774576260205627800573054, 6.14260017171135465786573044022
