| L(s) = 1 | + 4·4-s − 4·9-s − 4·11-s + 6·16-s − 32·23-s − 20·25-s − 16·36-s + 8·37-s − 16·44-s + 56·53-s + 24·67-s + 32·71-s − 128·92-s + 16·99-s − 80·100-s − 48·113-s + 8·121-s + 127-s + 131-s + 137-s + 139-s − 24·144-s + 32·148-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·4-s − 4/3·9-s − 1.20·11-s + 3/2·16-s − 6.67·23-s − 4·25-s − 8/3·36-s + 1.31·37-s − 2.41·44-s + 7.69·53-s + 2.93·67-s + 3.79·71-s − 13.3·92-s + 1.60·99-s − 8·100-s − 4.51·113-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2·144-s + 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0009904485432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0009904485432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 7 | \( 1 \) |
| 11 | \( ( 1 + 2 T + 2 T^{2} - 40 T^{3} - 161 T^{4} - 40 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| good | 3 | \( ( 1 + 2 T^{2} + 2 p T^{4} - 40 T^{6} - 113 T^{8} - 40 p^{2} T^{10} + 2 p^{5} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 5 | \( ( 1 + 2 p T^{2} + 46 T^{4} + 8 p T^{6} - 209 T^{8} + 8 p^{3} T^{10} + 46 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 42 T^{2} + 758 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 20 T^{2} + 111 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 - 42 T^{2} + 790 T^{4} - 10584 T^{6} + 181551 T^{8} - 10584 p^{2} T^{10} + 790 p^{4} T^{12} - 42 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{8} \) |
| 29 | \( ( 1 - 56 T^{2} + 1710 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 + 48 T^{2} + 1150 T^{4} - 36864 T^{6} - 1805949 T^{8} - 36864 p^{2} T^{10} + 1150 p^{4} T^{12} + 48 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - 2 T - 50 T^{2} + 40 T^{3} + 1399 T^{4} + 40 p T^{5} - 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 40 T^{2} + 2418 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 + 4 T^{2} + 2358 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 64 T^{2} - 2 T^{4} - 20480 T^{6} + 2865859 T^{8} - 20480 p^{2} T^{10} - 2 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 14 T + 62 T^{2} - 392 T^{3} + 4759 T^{4} - 392 p T^{5} + 62 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 226 T^{2} + 31366 T^{4} + 2881048 T^{6} + 198571183 T^{8} + 2881048 p^{2} T^{10} + 31366 p^{4} T^{12} + 226 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 154 T^{2} + 12046 T^{4} - 651112 T^{6} + 34095823 T^{8} - 651112 p^{2} T^{10} + 12046 p^{4} T^{12} - 154 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 6 T - 86 T^{2} + 72 T^{3} + 7983 T^{4} + 72 p T^{5} - 86 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 2 T + p T^{2} )^{16} \) |
| 73 | \( ( 1 + 8 T^{2} + 1486 T^{4} - 96640 T^{6} - 26717213 T^{8} - 96640 p^{2} T^{10} + 1486 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{4}( 1 + 131 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 82 T^{2} + 13758 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 + 82 T^{2} - 1197 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 - 60 T^{2} + 11318 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.52540698437774918240218203931, −2.46458192660685450404396363210, −2.46390388582458517720847239395, −2.28436597995334465448804068444, −2.22464372534298408366925681309, −2.21576874754179949364409360586, −2.12332518773302430259873419570, −2.09865761101782687371377965156, −2.01542668022905719310984436121, −1.86501967346130582687233585596, −1.83055492078967575354208415731, −1.79178438191096452732640328307, −1.59829082047027980874913969758, −1.47914465292805377906716923044, −1.39001034914917193001275771208, −1.35832860959836205056503349016, −1.23794370095875562241034010663, −1.23222461951027173771749299083, −0.812757489884072533201080216345, −0.78168537973840391630357107192, −0.66947157823741323286904419907, −0.42170164936454081909476076403, −0.41795811728474781162554094560, −0.35960209017697824235810842620, −0.00197134835585024933867475115,
0.00197134835585024933867475115, 0.35960209017697824235810842620, 0.41795811728474781162554094560, 0.42170164936454081909476076403, 0.66947157823741323286904419907, 0.78168537973840391630357107192, 0.812757489884072533201080216345, 1.23222461951027173771749299083, 1.23794370095875562241034010663, 1.35832860959836205056503349016, 1.39001034914917193001275771208, 1.47914465292805377906716923044, 1.59829082047027980874913969758, 1.79178438191096452732640328307, 1.83055492078967575354208415731, 1.86501967346130582687233585596, 2.01542668022905719310984436121, 2.09865761101782687371377965156, 2.12332518773302430259873419570, 2.21576874754179949364409360586, 2.22464372534298408366925681309, 2.28436597995334465448804068444, 2.46390388582458517720847239395, 2.46458192660685450404396363210, 2.52540698437774918240218203931
Plot not available for L-functions of degree greater than 10.
