| L(s) = 1 | + 3·5-s − 2·7-s + 5·9-s + 4·13-s + 5·25-s − 12·29-s − 6·35-s + 2·37-s + 15·45-s − 30·47-s − 9·49-s − 4·61-s − 10·63-s + 12·65-s + 16·67-s − 40·73-s + 4·79-s + 9·81-s + 12·83-s − 8·91-s − 4·97-s − 36·101-s + 20·117-s + 16·121-s + 18·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.755·7-s + 5/3·9-s + 1.10·13-s + 25-s − 2.22·29-s − 1.01·35-s + 0.328·37-s + 2.23·45-s − 4.37·47-s − 9/7·49-s − 0.512·61-s − 1.25·63-s + 1.48·65-s + 1.95·67-s − 4.68·73-s + 0.450·79-s + 81-s + 1.31·83-s − 0.838·91-s − 0.406·97-s − 3.58·101-s + 1.84·117-s + 1.45·121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.569593847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.569593847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 7 | $D_{4}$ | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 174 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 528 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - T - p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1926 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 3648 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 13950 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 109 T^{2} + 7896 T^{4} - 109 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 280 T^{2} + 34254 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.07827156611618828488171004361, −6.80526835972728989592724140279, −6.55624812140171852299232309750, −6.50322479138415080886145672269, −6.41320598889990220223492505210, −5.86986576893026860381060450440, −5.68574521851644921912655584642, −5.60728989479020817247624891464, −5.55808146181171028264368712793, −4.91730791073943375296738165090, −4.78106693676393916024196587797, −4.69993356775373143708741364011, −4.20053911965770776774727217881, −4.12570162011379962337667442254, −3.82547316460620827519997898517, −3.44160881432698790349330473059, −3.21494208131970797880474949107, −2.96044689980600488887318070184, −2.91871872017111522801500133503, −2.06532325746678494700357067828, −1.98325551657214501191908861200, −1.60934760200905005700198667783, −1.57502306626708735021521025029, −1.11085236737372056363683589779, −0.33514818386342813165710894177,
0.33514818386342813165710894177, 1.11085236737372056363683589779, 1.57502306626708735021521025029, 1.60934760200905005700198667783, 1.98325551657214501191908861200, 2.06532325746678494700357067828, 2.91871872017111522801500133503, 2.96044689980600488887318070184, 3.21494208131970797880474949107, 3.44160881432698790349330473059, 3.82547316460620827519997898517, 4.12570162011379962337667442254, 4.20053911965770776774727217881, 4.69993356775373143708741364011, 4.78106693676393916024196587797, 4.91730791073943375296738165090, 5.55808146181171028264368712793, 5.60728989479020817247624891464, 5.68574521851644921912655584642, 5.86986576893026860381060450440, 6.41320598889990220223492505210, 6.50322479138415080886145672269, 6.55624812140171852299232309750, 6.80526835972728989592724140279, 7.07827156611618828488171004361
