| L(s) = 1 | − 2·9-s − 2·13-s + 2·17-s − 4·25-s − 2·29-s + 2·37-s + 2·41-s − 2·49-s − 2·53-s − 2·61-s + 2·73-s + 3·81-s + 2·89-s + 2·97-s + 2·109-s + 2·113-s + 4·117-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·9-s − 2·13-s + 2·17-s − 4·25-s − 2·29-s + 2·37-s + 2·41-s − 2·49-s − 2·53-s − 2·61-s + 2·73-s + 3·81-s + 2·89-s + 2·97-s + 2·109-s + 2·113-s + 4·117-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7356255730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7356255730\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.70891248273647857718847515071, −6.16285076182293122390775343071, −6.09573983785060478883165724104, −6.08862588194168505814805882055, −5.99481906651498941146789773439, −5.67911402293675031802921425337, −5.65067382945860378773458271663, −5.05624440779986162453870070605, −5.04124535716138313403161523463, −4.94186402873202804668246052107, −4.77005920379001538220267151624, −4.15661976152353882462500083534, −4.05971169403785206166445338684, −3.97926708478563933588316304733, −3.66572330014829490973476375430, −3.16086895813228209228601510753, −2.99007783825929830520150373683, −2.98932195525673536259601004059, −2.96331479897636675510824701269, −2.00519150011210948246172494671, −2.00194846719238086705147570658, −1.95595052725420206781836907020, −1.85783863167669148192655382561, −0.68237076840298874323420351541, −0.61648077832895542661799122430,
0.61648077832895542661799122430, 0.68237076840298874323420351541, 1.85783863167669148192655382561, 1.95595052725420206781836907020, 2.00194846719238086705147570658, 2.00519150011210948246172494671, 2.96331479897636675510824701269, 2.98932195525673536259601004059, 2.99007783825929830520150373683, 3.16086895813228209228601510753, 3.66572330014829490973476375430, 3.97926708478563933588316304733, 4.05971169403785206166445338684, 4.15661976152353882462500083534, 4.77005920379001538220267151624, 4.94186402873202804668246052107, 5.04124535716138313403161523463, 5.05624440779986162453870070605, 5.65067382945860378773458271663, 5.67911402293675031802921425337, 5.99481906651498941146789773439, 6.08862588194168505814805882055, 6.09573983785060478883165724104, 6.16285076182293122390775343071, 6.70891248273647857718847515071
