| L(s) = 1 | − 24·9-s + 20·25-s + 4·29-s − 28·41-s + 18·49-s + 324·81-s + 68·101-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 8·9-s + 4·25-s + 0.742·29-s − 4.37·41-s + 18/7·49-s + 36·81-s + 6.76·101-s − 8·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06510761548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06510761548\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
| 23 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 + p T^{2} )^{8} \) |
| 7 | \( ( 1 - 9 T^{2} + 32 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + p T^{2} )^{8} \) |
| 13 | \( ( 1 - p T^{2} )^{8} \) |
| 17 | \( ( 1 + 11 T^{2} - 168 T^{4} + 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | \( ( 1 + 51 T^{2} + 1232 T^{4} + 51 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 - 101 T^{2} + 7392 T^{4} - 101 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2}( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + 111 T^{2} + 7832 T^{4} + 111 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2}( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 41 T^{2} - 5208 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 + 174 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−3.62539720177268157900757344410, −3.56936192007090404864714354983, −3.55728118899011783371262592912, −3.36359262749177654624770690561, −3.30676245461652783243884568126, −3.15088821386785695523983795958, −3.12659739086198694620414617399, −3.11670386896734681067569997534, −2.97332167050769695731187843206, −2.81794666121454283782123122131, −2.58132555990661226949379539085, −2.46231034709476134061237288727, −2.43061103322267684309924188724, −2.32974750975559208343373195804, −2.31235608551701654622644966725, −1.96457940962829690298066018271, −1.85031029663865860249947389488, −1.78237309686091182520245304873, −1.24964154119273593094300633008, −1.15501308559050647878910335618, −0.842081339616975730092438956600, −0.833040937079540689133397217535, −0.57236667146099631994767377791, −0.30772117322514140432805776825, −0.05081423824520979663742650248,
0.05081423824520979663742650248, 0.30772117322514140432805776825, 0.57236667146099631994767377791, 0.833040937079540689133397217535, 0.842081339616975730092438956600, 1.15501308559050647878910335618, 1.24964154119273593094300633008, 1.78237309686091182520245304873, 1.85031029663865860249947389488, 1.96457940962829690298066018271, 2.31235608551701654622644966725, 2.32974750975559208343373195804, 2.43061103322267684309924188724, 2.46231034709476134061237288727, 2.58132555990661226949379539085, 2.81794666121454283782123122131, 2.97332167050769695731187843206, 3.11670386896734681067569997534, 3.12659739086198694620414617399, 3.15088821386785695523983795958, 3.30676245461652783243884568126, 3.36359262749177654624770690561, 3.55728118899011783371262592912, 3.56936192007090404864714354983, 3.62539720177268157900757344410
Plot not available for L-functions of degree greater than 10.
