| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 3·9-s − 4·15-s − 8·17-s + 8·19-s − 4·21-s − 10·23-s + 25-s − 10·27-s − 6·29-s + 12·31-s + 4·35-s − 24·37-s + 10·41-s + 4·43-s + 6·45-s − 14·47-s + 9·49-s + 16·51-s − 8·53-s − 16·57-s − 12·59-s + 6·61-s + 6·63-s − 2·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s − 1.03·15-s − 1.94·17-s + 1.83·19-s − 0.872·21-s − 2.08·23-s + 1/5·25-s − 1.92·27-s − 1.11·29-s + 2.15·31-s + 0.676·35-s − 3.94·37-s + 1.56·41-s + 0.609·43-s + 0.894·45-s − 2.04·47-s + 9/7·49-s + 2.24·51-s − 1.09·53-s − 2.11·57-s − 1.56·59-s + 0.768·61-s + 0.755·63-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.162947349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162947349\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 10 p T^{5} - 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 10 T + 35 T^{2} + 190 T^{3} + 1396 T^{4} + 190 p T^{5} + 35 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T - 7 T^{2} - 90 T^{3} - 36 T^{4} - 90 p T^{5} - 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 70 T^{2} - 144 T^{3} + 51 T^{4} - 144 p T^{5} + 70 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 10 T + 17 T^{2} - 10 T^{3} + 1108 T^{4} - 10 p T^{5} + 17 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 22 T^{2} + 368 T^{3} - 2501 T^{4} + 368 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 14 T + 59 T^{2} + 602 T^{3} + 7348 T^{4} + 602 p T^{5} + 59 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 14 T^{2} + 144 T^{3} + 4923 T^{4} + 144 p T^{5} + 14 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T + 19 T^{2} - 298 T^{3} - 4532 T^{4} - 298 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T - 122 T^{2} - 80 T^{3} + 11539 T^{4} - 80 p T^{5} - 122 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 18 p T^{5} - 133 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 14 T + 131 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.37066044645943103308851492332, −7.33585201376107346540192324198, −6.87981915569236881268368140640, −6.83317749925184454572522298390, −6.44333503832701885315078760798, −6.41904495939661558822389766874, −5.95429030302776918113362440536, −5.69454031474100674333405374482, −5.68136778805612990920863920969, −5.39376855765798029680332070095, −5.34128921405890531874754445330, −4.63879131146943997356033840944, −4.62183148940079457118298043420, −4.53139349427312735920576175321, −4.23081924586108941434091526201, −3.81841909508918004462768781279, −3.47539106898845180001855828377, −3.12756383053117930732439469251, −3.03881253096132605292838542846, −2.26823031383768460065273737564, −1.97040975884848632450851485273, −1.86822181613396461112825376592, −1.68178210662869151364463014952, −0.962778779675070972594813309367, −0.34073225336231518616438204083,
0.34073225336231518616438204083, 0.962778779675070972594813309367, 1.68178210662869151364463014952, 1.86822181613396461112825376592, 1.97040975884848632450851485273, 2.26823031383768460065273737564, 3.03881253096132605292838542846, 3.12756383053117930732439469251, 3.47539106898845180001855828377, 3.81841909508918004462768781279, 4.23081924586108941434091526201, 4.53139349427312735920576175321, 4.62183148940079457118298043420, 4.63879131146943997356033840944, 5.34128921405890531874754445330, 5.39376855765798029680332070095, 5.68136778805612990920863920969, 5.69454031474100674333405374482, 5.95429030302776918113362440536, 6.41904495939661558822389766874, 6.44333503832701885315078760798, 6.83317749925184454572522298390, 6.87981915569236881268368140640, 7.33585201376107346540192324198, 7.37066044645943103308851492332
