L(s) = 1 | − 8·13-s − 16·17-s − 4·19-s + 8·29-s + 8·31-s + 24·37-s + 8·43-s + 4·49-s + 16·53-s − 24·59-s + 4·61-s + 8·67-s + 32·79-s + 16·83-s − 8·97-s + 8·101-s + 32·107-s + 12·109-s − 64·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 2.21·13-s − 3.88·17-s − 0.917·19-s + 1.48·29-s + 1.43·31-s + 3.94·37-s + 1.21·43-s + 4/7·49-s + 2.19·53-s − 3.12·59-s + 0.512·61-s + 0.977·67-s + 3.60·79-s + 1.75·83-s − 0.812·97-s + 0.796·101-s + 3.09·107-s + 1.14·109-s − 6.02·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1760977835\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1760977835\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
good | 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 20 T^{3} - 146 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1714 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 264 T^{3} + 2162 T^{4} - 264 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2520 T^{3} + 17426 T^{4} - 2520 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} + 104 T^{3} - 2798 T^{4} + 104 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 3048 T^{3} + 27634 T^{4} + 3048 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 324 T^{3} - 7042 T^{4} + 324 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 88 T^{3} - 2894 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 19910 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 240 T^{3} - 4174 T^{4} - 240 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 284 T^{2} + 35494 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.35606793983371324300846224081, −6.33028208385152991339262919303, −5.94023234214984458584030039420, −5.55004075624154495439491562336, −5.25881038360730769682031326321, −5.08799199223083203900548798445, −4.97135533047448856814342303737, −4.69235330165107981670743759515, −4.63084691622531435835797282866, −4.35477739940796731533644526514, −4.17484155483564348684506678060, −4.06786184032937518074176938760, −3.77387298925688915157723342194, −3.68255788747597749968885381143, −2.89424820699942022229478186582, −2.83439520183856317711521675836, −2.62920416417805850310614101208, −2.42924184599820700324495581500, −2.38477511088773178381444526182, −2.17349349298524991554505648708, −1.88475167496763988928312597648, −1.26912388029076335043821907003, −0.896957231618696696872705085188, −0.72649767875707014177540689048, −0.07657363644073475144080116209,
0.07657363644073475144080116209, 0.72649767875707014177540689048, 0.896957231618696696872705085188, 1.26912388029076335043821907003, 1.88475167496763988928312597648, 2.17349349298524991554505648708, 2.38477511088773178381444526182, 2.42924184599820700324495581500, 2.62920416417805850310614101208, 2.83439520183856317711521675836, 2.89424820699942022229478186582, 3.68255788747597749968885381143, 3.77387298925688915157723342194, 4.06786184032937518074176938760, 4.17484155483564348684506678060, 4.35477739940796731533644526514, 4.63084691622531435835797282866, 4.69235330165107981670743759515, 4.97135533047448856814342303737, 5.08799199223083203900548798445, 5.25881038360730769682031326321, 5.55004075624154495439491562336, 5.94023234214984458584030039420, 6.33028208385152991339262919303, 6.35606793983371324300846224081