| L(s) = 1 | − 4·3-s + 6·9-s − 8·13-s + 32·39-s − 15·81-s − 48·117-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 4·3-s + 6·9-s − 8·13-s + 32·39-s − 15·81-s − 48·117-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02244312037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02244312037\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{4} \) |
| 7 | \( 1 \) |
| 13 | \( ( 1 + T )^{8} \) |
| good | 2 | \( 1 - T^{8} + T^{16} \) |
| 5 | \( 1 - T^{8} + T^{16} \) |
| 11 | \( 1 - T^{8} + T^{16} \) |
| 17 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 19 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 23 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 29 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 31 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 37 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 41 | \( ( 1 + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{2} )^{8} \) |
| 47 | \( 1 - T^{8} + T^{16} \) |
| 53 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 59 | \( 1 - T^{8} + T^{16} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 67 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 71 | \( ( 1 + T^{8} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 79 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( 1 - T^{8} + T^{16} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
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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
−4.16973415007004942936615968804, −4.02617798134232987566453810397, −4.02229305405552366948908190894, −3.98841061880335035524970260228, −3.79875805411527859739085903414, −3.33089529554139247745752495567, −3.22116140259266537153710783466, −3.02338338272975464266054503340, −2.98536851486926970412693212717, −2.92375765469683277407709145621, −2.88020834224638559569301326556, −2.80085278337448965096891717605, −2.74217277227607786957084605597, −2.29521423180542013200533193948, −2.20931331483219446306857925897, −2.09518598820693393259355413120, −2.05611580767749605967054170809, −2.01993222046230178137897736561, −1.86739056322196572454119861399, −1.39170354451098520297033575112, −1.11076249765841372462635910664, −1.02268019764110511270924881952, −0.73179004147242198548520352878, −0.39843003109222045745526984364, −0.22898205068513619376748831304,
0.22898205068513619376748831304, 0.39843003109222045745526984364, 0.73179004147242198548520352878, 1.02268019764110511270924881952, 1.11076249765841372462635910664, 1.39170354451098520297033575112, 1.86739056322196572454119861399, 2.01993222046230178137897736561, 2.05611580767749605967054170809, 2.09518598820693393259355413120, 2.20931331483219446306857925897, 2.29521423180542013200533193948, 2.74217277227607786957084605597, 2.80085278337448965096891717605, 2.88020834224638559569301326556, 2.92375765469683277407709145621, 2.98536851486926970412693212717, 3.02338338272975464266054503340, 3.22116140259266537153710783466, 3.33089529554139247745752495567, 3.79875805411527859739085903414, 3.98841061880335035524970260228, 4.02229305405552366948908190894, 4.02617798134232987566453810397, 4.16973415007004942936615968804
Plot not available for L-functions of degree greater than 10.
