| L(s) = 1 | + 4·5-s + 4·7-s − 2·9-s − 4·11-s + 2·13-s + 8·17-s − 6·19-s + 6·23-s + 10·25-s + 2·27-s + 8·29-s − 4·31-s + 16·35-s + 10·37-s + 12·41-s − 2·43-s − 8·45-s + 6·47-s + 10·49-s + 6·53-s − 16·55-s − 4·59-s + 26·61-s − 8·63-s + 8·65-s + 10·67-s − 4·71-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.51·7-s − 2/3·9-s − 1.20·11-s + 0.554·13-s + 1.94·17-s − 1.37·19-s + 1.25·23-s + 2·25-s + 0.384·27-s + 1.48·29-s − 0.718·31-s + 2.70·35-s + 1.64·37-s + 1.87·41-s − 0.304·43-s − 1.19·45-s + 0.875·47-s + 10/7·49-s + 0.824·53-s − 2.15·55-s − 0.520·59-s + 3.32·61-s − 1.00·63-s + 0.992·65-s + 1.22·67-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(21.43596732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.43596732\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 2 T^{2} - 2 T^{3} - 2 T^{4} - 2 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 2 T + 18 T^{2} - 80 T^{3} + 226 T^{4} - 80 p T^{5} + 18 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 8 T + 82 T^{2} - 398 T^{3} + 2186 T^{4} - 398 p T^{5} + 82 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 6 T + 48 T^{2} + 10 p T^{3} + 1006 T^{4} + 10 p^{2} T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 6 T + 36 T^{2} - 178 T^{3} + 18 p T^{4} - 178 p T^{5} + 36 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 84 T^{2} + 238 T^{3} + 3462 T^{4} + 238 p T^{5} + 84 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 10 T + 116 T^{2} - 866 T^{3} + 5582 T^{4} - 866 p T^{5} + 116 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 12 T + 160 T^{2} - 1106 T^{3} + 9090 T^{4} - 1106 p T^{5} + 160 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 2 T + 100 T^{2} + 454 T^{3} + 4758 T^{4} + 454 p T^{5} + 100 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 6 T + 118 T^{2} - 688 T^{3} + 7718 T^{4} - 688 p T^{5} + 118 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 6 T + 68 T^{2} - 590 T^{3} + 3198 T^{4} - 590 p T^{5} + 68 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 4 T + 128 T^{2} - 102 T^{3} + 6694 T^{4} - 102 p T^{5} + 128 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 26 T + 384 T^{2} - 4020 T^{3} + 33770 T^{4} - 4020 p T^{5} + 384 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 10 T + 232 T^{2} - 1974 T^{3} + 22102 T^{4} - 1974 p T^{5} + 232 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 4 T + 92 T^{2} + 468 T^{3} + 11686 T^{4} + 468 p T^{5} + 92 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 24 T + 498 T^{2} - 6002 T^{3} + 63002 T^{4} - 6002 p T^{5} + 498 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 8 T + 36 T^{2} - 12 T^{3} + 3734 T^{4} - 12 p T^{5} + 36 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2 T + 72 T^{2} - 10 T^{3} + 2110 T^{4} - 10 p T^{5} + 72 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 6 T + 196 T^{2} + 1250 T^{3} + 21638 T^{4} + 1250 p T^{5} + 196 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2 T + 340 T^{2} + 534 T^{3} + 47686 T^{4} + 534 p T^{5} + 340 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.74981687156121929725538737935, −5.47346421047166254754131141281, −5.23543183734742092654351753121, −5.18037586049611515335963419902, −4.97874729555542849929508870536, −4.81571417891431249242479670603, −4.64481562519406027074480679096, −4.40197609676264434402413910551, −4.28182893112421310592387194632, −3.83609726191291898158509463084, −3.67890115305164753701206912898, −3.51777634287646554684628261029, −3.41735495456790402077140998510, −2.93037725275282670420591751759, −2.71413234508814433752411868360, −2.52377283050528291484489036829, −2.50993271048263645515647090420, −2.25775600067394010959395928518, −2.05576996891357544651854160462, −1.79855385240207087440330348550, −1.43760723339485176890389206414, −1.11862314056325170889941905061, −0.904499328125522667844029867054, −0.74829867617094506966676320528, −0.54498323475716917015383401219,
0.54498323475716917015383401219, 0.74829867617094506966676320528, 0.904499328125522667844029867054, 1.11862314056325170889941905061, 1.43760723339485176890389206414, 1.79855385240207087440330348550, 2.05576996891357544651854160462, 2.25775600067394010959395928518, 2.50993271048263645515647090420, 2.52377283050528291484489036829, 2.71413234508814433752411868360, 2.93037725275282670420591751759, 3.41735495456790402077140998510, 3.51777634287646554684628261029, 3.67890115305164753701206912898, 3.83609726191291898158509463084, 4.28182893112421310592387194632, 4.40197609676264434402413910551, 4.64481562519406027074480679096, 4.81571417891431249242479670603, 4.97874729555542849929508870536, 5.18037586049611515335963419902, 5.23543183734742092654351753121, 5.47346421047166254754131141281, 5.74981687156121929725538737935
