| L(s) = 1 | + 3·5-s + 2·7-s − 3·11-s − 4·13-s + 6·17-s − 20·19-s + 9·23-s + 4·25-s − 6·29-s + 4·31-s + 6·35-s + 8·37-s + 15·41-s + 43-s + 49-s + 12·53-s − 9·55-s + 3·59-s + 11·61-s − 12·65-s + 13·67-s + 6·71-s + 14·73-s − 6·77-s + 7·79-s − 12·83-s + 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.755·7-s − 0.904·11-s − 1.10·13-s + 1.45·17-s − 4.58·19-s + 1.87·23-s + 4/5·25-s − 1.11·29-s + 0.718·31-s + 1.01·35-s + 1.31·37-s + 2.34·41-s + 0.152·43-s + 1/7·49-s + 1.64·53-s − 1.21·55-s + 0.390·59-s + 1.40·61-s − 1.48·65-s + 1.58·67-s + 0.712·71-s + 1.63·73-s − 0.683·77-s + 0.787·79-s − 1.31·83-s + 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.393319930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.393319930\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 144 T^{3} - 729 T^{4} - 144 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 720 p T^{5} + 95 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - T - 11 T^{2} + 74 T^{3} - 1748 T^{4} + 74 p T^{5} - 11 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3 T - 37 T^{2} + 216 T^{3} - 1896 T^{4} + 216 p T^{5} - 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 484 p T^{5} + 43 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 7 T - 47 T^{2} + 434 T^{3} - 896 T^{4} + 434 p T^{5} - 47 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 74 T^{2} - 1152 T^{3} - 13941 T^{4} - 1152 p T^{5} + 74 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 74 p T^{5} - 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.01939499976104844125373286209, −6.01127439913091472581620783693, −5.76952230259523669293241368473, −5.41407055385342249534684894377, −5.41005469461898765112348707028, −5.37544098127678125316511965914, −4.94720024382252209794702424066, −4.82309900193697073488603916653, −4.55966097461479450949064063387, −4.16512311095490742439285629334, −4.08217429591477220010863972869, −4.08166409229304674079062052878, −4.00507848204479514219649128249, −3.19980676095244443743067473082, −3.02919448448727841338515452458, −3.01766239396072143705825950230, −2.54096353814137743253971887666, −2.37273876019710440250911096460, −2.26582046524676467928483513536, −1.95846562447836578476529722861, −1.85866009529116106074133571208, −1.52208907944074665117997098941, −0.818426300375366647554410350006, −0.72877027957706749262282104614, −0.49076874362726280204483575554,
0.49076874362726280204483575554, 0.72877027957706749262282104614, 0.818426300375366647554410350006, 1.52208907944074665117997098941, 1.85866009529116106074133571208, 1.95846562447836578476529722861, 2.26582046524676467928483513536, 2.37273876019710440250911096460, 2.54096353814137743253971887666, 3.01766239396072143705825950230, 3.02919448448727841338515452458, 3.19980676095244443743067473082, 4.00507848204479514219649128249, 4.08166409229304674079062052878, 4.08217429591477220010863972869, 4.16512311095490742439285629334, 4.55966097461479450949064063387, 4.82309900193697073488603916653, 4.94720024382252209794702424066, 5.37544098127678125316511965914, 5.41005469461898765112348707028, 5.41407055385342249534684894377, 5.76952230259523669293241368473, 6.01127439913091472581620783693, 6.01939499976104844125373286209
