| L(s) = 1 | + 8·23-s − 2·49-s − 2·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 8·23-s − 2·49-s − 2·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.035598821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.035598821\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{8} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.36572528731631120153281003542, −6.31905718175424271463082853833, −6.23746786558111423079833758947, −5.59803825940245193684963537792, −5.38873514127380706061955727988, −5.37920740541460559824249463172, −5.09557412397718818382599342686, −4.99963572458858540318388140270, −4.89391953767978279482845126036, −4.79089158383282646471957162967, −4.33012411136294082458597471480, −4.10721004759174667684788453529, −3.92916743748449040022177749409, −3.61305270898659632363975126260, −3.36273230211768659078341144150, −3.10666737361676896194971870614, −2.88441013950534007533070264167, −2.76511523523130414000747269851, −2.65681194409379412654779901492, −2.49101923101141950941641141133, −1.68251305297090407784206146699, −1.53715507072371589536864120797, −1.23206927424380995611394190655, −1.14267381849883199594566833199, −0.66417315727420411801702924684,
0.66417315727420411801702924684, 1.14267381849883199594566833199, 1.23206927424380995611394190655, 1.53715507072371589536864120797, 1.68251305297090407784206146699, 2.49101923101141950941641141133, 2.65681194409379412654779901492, 2.76511523523130414000747269851, 2.88441013950534007533070264167, 3.10666737361676896194971870614, 3.36273230211768659078341144150, 3.61305270898659632363975126260, 3.92916743748449040022177749409, 4.10721004759174667684788453529, 4.33012411136294082458597471480, 4.79089158383282646471957162967, 4.89391953767978279482845126036, 4.99963572458858540318388140270, 5.09557412397718818382599342686, 5.37920740541460559824249463172, 5.38873514127380706061955727988, 5.59803825940245193684963537792, 6.23746786558111423079833758947, 6.31905718175424271463082853833, 6.36572528731631120153281003542
