| L(s) = 1 | + 2·4-s + 4·9-s + 16-s − 40·19-s − 8·31-s + 8·36-s − 36·41-s − 8·49-s − 12·59-s − 32·61-s − 2·64-s + 48·71-s − 80·76-s + 8·79-s − 6·81-s + 72·89-s − 24·101-s − 64·109-s − 4·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s + 4/3·9-s + 1/4·16-s − 9.17·19-s − 1.43·31-s + 4/3·36-s − 5.62·41-s − 8/7·49-s − 1.56·59-s − 4.09·61-s − 1/4·64-s + 5.69·71-s − 9.17·76-s + 0.900·79-s − 2/3·81-s + 7.63·89-s − 2.38·101-s − 6.13·109-s − 0.363·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2280216460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2280216460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8 T^{2} + 46 T^{4} - 640 T^{6} - 5213 T^{8} - 640 p^{2} T^{10} + 46 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 + 32 T^{2} + 526 T^{4} + 5120 T^{6} + 46387 T^{8} + 5120 p^{2} T^{10} + 526 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 40 T^{2} + 1071 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 8 p T^{5} - 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T^{2} - 702 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 110 T^{2} + 139 p T^{4} + 266750 T^{6} + 11699428 T^{8} + 266750 p^{2} T^{10} + 139 p^{5} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 68 T^{2} + 2506 T^{4} - 156400 T^{6} - 10608173 T^{8} - 156400 p^{2} T^{10} + 2506 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 128 T^{2} + 8850 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 6 T - 85 T^{2} + 18 T^{3} + 9036 T^{4} + 18 p T^{5} - 85 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 71 | \( ( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 4 T + 70 T^{2} + 848 T^{3} - 4589 T^{4} + 848 p T^{5} + 70 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 302 T^{2} + 54841 T^{4} + 6820670 T^{6} + 646505092 T^{8} + 6820670 p^{2} T^{10} + 54841 p^{4} T^{12} + 302 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 9 T + p T^{2} )^{8} \) |
| 97 | \( 1 + 2 p T^{2} + 9793 T^{4} + 18050 p T^{6} + 332586244 T^{8} + 18050 p^{3} T^{10} + 9793 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) |
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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
−4.87666415536503729289740062123, −4.82350974321019474402189743799, −4.77773364033562181998256893815, −4.41506999650727758557990557785, −4.18290591960772263574508911061, −4.13718781182922048881002259725, −3.96457159041005449739731872180, −3.92721899495613422447558742065, −3.90533353349964947421756984564, −3.59583320787432508317988168847, −3.51645064183507957150406719971, −3.36833338234159600190892725391, −3.05200336641778363831110661705, −2.74894366929963217297587744278, −2.67407650321425084919376402984, −2.38060137942445477817699916034, −2.33110614162747710902015764960, −2.16337020735023492830108763976, −1.93547569987324542925252039391, −1.81270793279332924964849194958, −1.62903544506915352212586937096, −1.55997885681308033660358949425, −1.41287199120117231602451667971, −0.36335831074920980610730063109, −0.13875971057856137959227205614,
0.13875971057856137959227205614, 0.36335831074920980610730063109, 1.41287199120117231602451667971, 1.55997885681308033660358949425, 1.62903544506915352212586937096, 1.81270793279332924964849194958, 1.93547569987324542925252039391, 2.16337020735023492830108763976, 2.33110614162747710902015764960, 2.38060137942445477817699916034, 2.67407650321425084919376402984, 2.74894366929963217297587744278, 3.05200336641778363831110661705, 3.36833338234159600190892725391, 3.51645064183507957150406719971, 3.59583320787432508317988168847, 3.90533353349964947421756984564, 3.92721899495613422447558742065, 3.96457159041005449739731872180, 4.13718781182922048881002259725, 4.18290591960772263574508911061, 4.41506999650727758557990557785, 4.77773364033562181998256893815, 4.82350974321019474402189743799, 4.87666415536503729289740062123
Plot not available for L-functions of degree greater than 10.
