| L(s) = 1 | − 8·3-s + 32·9-s − 8·13-s − 2·16-s + 4·25-s − 88·27-s + 64·39-s − 16·43-s + 16·48-s − 32·61-s − 48·67-s + 24·73-s − 32·75-s + 206·81-s + 16·97-s − 48·103-s − 256·117-s − 40·121-s + 127-s + 128·129-s + 131-s + 137-s + 139-s − 64·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 32/3·9-s − 2.21·13-s − 1/2·16-s + 4/5·25-s − 16.9·27-s + 10.2·39-s − 2.43·43-s + 2.30·48-s − 4.09·61-s − 5.86·67-s + 2.80·73-s − 3.69·75-s + 22.8·81-s + 1.62·97-s − 4.72·103-s − 23.6·117-s − 3.63·121-s + 0.0887·127-s + 11.2·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.33·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02017558202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02017558202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 + T^{4} )^{2} \) |
| good | 7 | \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 4 T + 8 T^{2} + 12 T^{3} - 82 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 178 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 28 T^{2} + 342 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 1106 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 8 T + 32 T^{2} + 24 T^{3} - 1582 T^{4} + 24 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 644 T^{4} + 3172230 T^{8} + 644 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 + 4828 T^{4} + 13645734 T^{8} + 4828 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 136 T^{2} + 11202 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 24 T + 288 T^{2} + 2184 T^{3} + 15986 T^{4} + 2184 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 32 T^{2} + 2562 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 12 T + 72 T^{2} - 516 T^{3} + 2798 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 296 T^{2} + 34290 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 12466 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 268 T^{2} + 32262 T^{4} + 268 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 8 T + 32 T^{2} - 744 T^{3} + 17282 T^{4} - 744 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.70065284424148711728586149022, −4.58654977757880897072969566590, −4.51108066876830105169121974798, −4.43861582749252643503474769174, −4.20859796228025181840564426864, −3.74620680755994955245182727496, −3.72331903374651836002115399120, −3.71465378532330843198774169791, −3.66168473135700658217784601471, −3.41805821829252120352112090655, −3.01827393744987360127559531679, −2.92749906540372938932034199923, −2.75950670067094563259586136731, −2.74252274766563989410714641979, −2.37235773137567227846136848616, −2.33255341142453757616483201392, −2.20774796845626966531739897226, −1.58628915072729756364566966460, −1.54938482530216525334286388301, −1.46267551943397379036372349319, −1.43396195901344587719508504781, −1.08388112414309498813505700366, −0.57248737610152807230572315066, −0.23432258351520421328157749539, −0.12463525596437182630034052012,
0.12463525596437182630034052012, 0.23432258351520421328157749539, 0.57248737610152807230572315066, 1.08388112414309498813505700366, 1.43396195901344587719508504781, 1.46267551943397379036372349319, 1.54938482530216525334286388301, 1.58628915072729756364566966460, 2.20774796845626966531739897226, 2.33255341142453757616483201392, 2.37235773137567227846136848616, 2.74252274766563989410714641979, 2.75950670067094563259586136731, 2.92749906540372938932034199923, 3.01827393744987360127559531679, 3.41805821829252120352112090655, 3.66168473135700658217784601471, 3.71465378532330843198774169791, 3.72331903374651836002115399120, 3.74620680755994955245182727496, 4.20859796228025181840564426864, 4.43861582749252643503474769174, 4.51108066876830105169121974798, 4.58654977757880897072969566590, 4.70065284424148711728586149022
Plot not available for L-functions of degree greater than 10.
