| L(s) = 1 | + 4-s + 4·13-s + 16-s + 26·25-s − 8·37-s + 38·49-s + 4·52-s + 4·61-s + 5·64-s + 4·73-s − 8·97-s + 26·100-s + 16·109-s − 64·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 62·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.10·13-s + 1/4·16-s + 26/5·25-s − 1.31·37-s + 38/7·49-s + 0.554·52-s + 0.512·61-s + 5/8·64-s + 0.468·73-s − 0.812·97-s + 13/5·100-s + 1.53·109-s − 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.76·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.368253559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.368253559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 13 T^{2} + 84 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 19 T^{2} + 180 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 32 T^{2} + 465 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 61 T^{2} + 1500 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 49 T^{2} + 1248 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 77 T^{2} + 2532 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 109 T^{2} + 4644 T^{4} - 109 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 55 T^{2} + 1680 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 118 T^{2} + 6315 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 64 T^{2} + 4689 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 53 T^{2} + 756 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 136 T^{2} + 9054 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 56 T^{2} + 3753 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - T + 114 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 160 T^{2} + 15345 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 140 T^{2} + 10230 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - T + 138 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 115 T^{2} + 15780 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 221 T^{2} + 25980 T^{4} + 221 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 184 T^{2} + 21006 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 2 T + 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−5.27564601599521067790283106462, −5.23515222780875294334058602536, −4.93323354434892314512348363521, −4.73248921665591412137331898598, −4.67753710928310399042654080518, −4.54193886022331215679694501071, −4.14367620601738205677447143481, −4.01669742209851357022854295658, −4.01039671192451859500182714473, −3.90266799760725061837440956758, −3.65806377405000152538572059586, −3.49744917356767410903429624991, −3.28600803525524826809115520290, −3.10651977497376665175931290415, −2.89631727969944702863485228412, −2.70098056416578118673765642794, −2.62674774327913853860844814118, −2.41528668247720303456925436152, −2.32008891818036613307317260837, −2.04904739439121086414643269009, −1.53111853418433834190089795416, −1.37286885085961951782937017533, −1.20363598546546291373557786315, −0.897553060799138574777162223210, −0.64195917548714101357827461841,
0.64195917548714101357827461841, 0.897553060799138574777162223210, 1.20363598546546291373557786315, 1.37286885085961951782937017533, 1.53111853418433834190089795416, 2.04904739439121086414643269009, 2.32008891818036613307317260837, 2.41528668247720303456925436152, 2.62674774327913853860844814118, 2.70098056416578118673765642794, 2.89631727969944702863485228412, 3.10651977497376665175931290415, 3.28600803525524826809115520290, 3.49744917356767410903429624991, 3.65806377405000152538572059586, 3.90266799760725061837440956758, 4.01039671192451859500182714473, 4.01669742209851357022854295658, 4.14367620601738205677447143481, 4.54193886022331215679694501071, 4.67753710928310399042654080518, 4.73248921665591412137331898598, 4.93323354434892314512348363521, 5.23515222780875294334058602536, 5.27564601599521067790283106462
Plot not available for L-functions of degree greater than 10.
