| L(s) = 1 | + 4-s + 25-s + 2·49-s + 4·61-s − 64-s + 100-s + 8·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4-s + 25-s + 2·49-s + 4·61-s − 64-s + 100-s + 8·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.705725655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.705725655\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−7.04057901194451708312205217036, −6.54963811539873896149969850818, −6.54032462287971348434811516737, −6.36349885480488703167579885362, −5.94122735338285999926038035738, −5.87940670171164686388235316514, −5.83014395926782614067774759010, −5.31831640933861909176935274380, −5.17139285300270339867545575216, −4.99548007048139648599869820669, −4.78424812946469498178519331162, −4.39507910584475596076420157303, −4.36669788582558019045778846385, −3.91024108092007094341439498897, −3.76447165701797227382057503573, −3.39895092753292895614691590449, −3.32885454178301273009373594473, −3.04657426948438142028518282849, −2.48602737220232797426061366771, −2.41088543567174479381565954539, −2.23628494430828299140949990625, −2.06023936028566122535163056463, −1.51862118490874178132704767725, −0.975806822276998642897698751818, −0.925882931185934536573689556123,
0.925882931185934536573689556123, 0.975806822276998642897698751818, 1.51862118490874178132704767725, 2.06023936028566122535163056463, 2.23628494430828299140949990625, 2.41088543567174479381565954539, 2.48602737220232797426061366771, 3.04657426948438142028518282849, 3.32885454178301273009373594473, 3.39895092753292895614691590449, 3.76447165701797227382057503573, 3.91024108092007094341439498897, 4.36669788582558019045778846385, 4.39507910584475596076420157303, 4.78424812946469498178519331162, 4.99548007048139648599869820669, 5.17139285300270339867545575216, 5.31831640933861909176935274380, 5.83014395926782614067774759010, 5.87940670171164686388235316514, 5.94122735338285999926038035738, 6.36349885480488703167579885362, 6.54032462287971348434811516737, 6.54963811539873896149969850818, 7.04057901194451708312205217036
