| L(s) = 1 | + 48·9-s + 144·17-s − 32·25-s + 80·41-s − 28·49-s − 464·73-s + 1.18e3·81-s − 624·89-s − 272·97-s − 608·113-s + 616·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 6.91e3·153-s + 157-s + 163-s + 167-s − 1.18e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 16/3·9-s + 8.47·17-s − 1.27·25-s + 1.95·41-s − 4/7·49-s − 6.35·73-s + 14.5·81-s − 7.01·89-s − 2.80·97-s − 5.38·113-s + 5.09·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 45.1·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 7.00·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.658977986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658977986\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 - 8 p T^{2} + 274 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 5 | \( ( 1 + 16 T^{2} - 254 T^{4} + 16 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 28 p T^{2} + 48390 T^{4} - 28 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 592 T^{2} + 143170 T^{4} + 592 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 36 T + 870 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 8 T^{2} + 249106 T^{4} + 8 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 268 T^{2} + 477286 T^{4} - 268 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 1178 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1380 T^{2} + 1419974 T^{4} - 1380 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 1780 T^{2} + 2031622 T^{4} + 1780 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 20 T + 3174 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 6580 T^{2} + 17648902 T^{4} - 6580 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 7492 T^{2} + 23390470 T^{4} - 7492 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 1604 T^{2} - 6155034 T^{4} + 1604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 9208 T^{2} + 43383250 T^{4} - 9208 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 13232 T^{2} + 71110338 T^{4} + 13232 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10660 T^{2} + 62288614 T^{4} - 10660 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 9412 T^{2} + 47279686 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 116 T + 13894 T^{2} + 116 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 16900 T^{2} + 142880134 T^{4} - 16900 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 25912 T^{2} + 262120210 T^{4} - 25912 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 156 T + 20774 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 68 T + 3046 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.68482930758597084771049067155, −3.65710603866805242958524059280, −3.62020619878021606850755790448, −3.32209229748167397074215825033, −3.19711129858355616798178671284, −3.01239284040672893321199804733, −2.93038123619989342506128500219, −2.91969252473253601653920045442, −2.77976745193571899978394696516, −2.72058826377050869549971784177, −2.42600716547161018002246118532, −2.17834063630040463896332217730, −1.99782309880377823547633040802, −1.78311234396837094631928787505, −1.77003762073634402241237114159, −1.42500387626336305613909958200, −1.35712191247542349415635776299, −1.24207954382935543286620492625, −1.20820512884525231893507062711, −1.16936776573594438803545203241, −1.15053616377806394726625332096, −1.12614105526621301218561957680, −0.63638037359480760569348258316, −0.35731015220892620143535767457, −0.04148731970417866683723149510,
0.04148731970417866683723149510, 0.35731015220892620143535767457, 0.63638037359480760569348258316, 1.12614105526621301218561957680, 1.15053616377806394726625332096, 1.16936776573594438803545203241, 1.20820512884525231893507062711, 1.24207954382935543286620492625, 1.35712191247542349415635776299, 1.42500387626336305613909958200, 1.77003762073634402241237114159, 1.78311234396837094631928787505, 1.99782309880377823547633040802, 2.17834063630040463896332217730, 2.42600716547161018002246118532, 2.72058826377050869549971784177, 2.77976745193571899978394696516, 2.91969252473253601653920045442, 2.93038123619989342506128500219, 3.01239284040672893321199804733, 3.19711129858355616798178671284, 3.32209229748167397074215825033, 3.62020619878021606850755790448, 3.65710603866805242958524059280, 3.68482930758597084771049067155
Plot not available for L-functions of degree greater than 10.
