| L(s) = 1 | + 20·7-s − 40·13-s − 23·16-s + 40·25-s − 32·31-s + 80·37-s − 40·43-s + 200·49-s + 232·61-s − 280·67-s + 440·73-s − 800·91-s + 20·97-s + 140·103-s − 460·112-s − 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 800·169-s + 173-s + ⋯ |
| L(s) = 1 | + 20/7·7-s − 3.07·13-s − 1.43·16-s + 8/5·25-s − 1.03·31-s + 2.16·37-s − 0.930·43-s + 4.08·49-s + 3.80·61-s − 4.17·67-s + 6.02·73-s − 8.79·91-s + 0.206·97-s + 1.35·103-s − 4.10·112-s − 0.132·121-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4.73·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(17.22982531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.22982531\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 8 p T^{2} + 39 p^{2} T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} \) |
| good | 2 | \( 1 + 23 T^{4} + 273 T^{8} + 23 p^{8} T^{12} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 10 T + 50 T^{2} + 480 T^{3} - 4801 T^{4} + 480 p^{2} T^{5} + 50 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 8 T^{2} - 14577 T^{4} + 8 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 20 T + 200 T^{2} - 2760 T^{3} - 56161 T^{4} - 2760 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 144322 T^{4} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 398 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 - 517762 T^{4} + 189766503363 T^{8} - 517762 p^{8} T^{12} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 568 T^{2} - 384657 T^{4} - 568 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 8 T - 897 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 20 T + 200 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 2362 T^{2} + 2753283 T^{4} - 2362 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 20 T + 200 T^{2} - 69960 T^{3} - 4118401 T^{4} - 69960 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 + 8681918 T^{4} + 51564413496963 T^{8} + 8681918 p^{8} T^{12} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 3037282 T^{4} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 4712 T^{2} + 10085583 T^{4} + 4712 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 58 T - 357 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 + 140 T + 9800 T^{2} + 115080 T^{3} - 12095521 T^{4} + 115080 p^{2} T^{5} + 9800 p^{4} T^{6} + 140 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 6082 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 110 T + 6050 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + 12338 T^{2} + 113276163 T^{4} + 12338 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 30948638 T^{4} - 1294474038083997 T^{8} + 30948638 p^{8} T^{12} + p^{16} T^{16} \) |
| 89 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 97 | \( ( 1 - 10 T + 50 T^{2} + 187680 T^{3} - 89467681 T^{4} + 187680 p^{2} T^{5} + 50 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.64897182936310250161074030089, −4.54641298182923400719600449977, −4.46968934274527551023347130824, −4.46681186747022836630959840992, −4.18783359091639900176481467873, −4.10581530603600100404992290225, −4.01277910219000292246981813985, −3.93024760374787806435654056782, −3.30051113190840078345788710464, −3.28799782025716302389956245667, −3.19958990164059556225284551991, −3.04842711188112782948636152664, −2.74751683014254781328319201531, −2.73108853449954190467969906399, −2.41338314048273461268797627623, −2.17756299101604149336355291148, −2.05536650468224523762537964178, −2.00053678896078430869242214178, −1.84061328904944644722237138066, −1.60851625053961948931938487570, −1.51740588257524858104384594683, −0.72917276988044757311693842144, −0.62725646656248607109211676289, −0.62317473013916487506518725824, −0.54715903863425491032257209176,
0.54715903863425491032257209176, 0.62317473013916487506518725824, 0.62725646656248607109211676289, 0.72917276988044757311693842144, 1.51740588257524858104384594683, 1.60851625053961948931938487570, 1.84061328904944644722237138066, 2.00053678896078430869242214178, 2.05536650468224523762537964178, 2.17756299101604149336355291148, 2.41338314048273461268797627623, 2.73108853449954190467969906399, 2.74751683014254781328319201531, 3.04842711188112782948636152664, 3.19958990164059556225284551991, 3.28799782025716302389956245667, 3.30051113190840078345788710464, 3.93024760374787806435654056782, 4.01277910219000292246981813985, 4.10581530603600100404992290225, 4.18783359091639900176481467873, 4.46681186747022836630959840992, 4.46968934274527551023347130824, 4.54641298182923400719600449977, 4.64897182936310250161074030089
Plot not available for L-functions of degree greater than 10.
