| L(s) = 1 | + 2·5-s + 9-s + 4·17-s + 4·19-s − 2·23-s + 25-s + 2·43-s + 2·45-s + 2·59-s − 2·61-s − 4·71-s − 2·79-s + 2·83-s + 8·85-s − 4·89-s + 8·95-s − 2·97-s − 4·103-s + 4·109-s − 4·115-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·5-s + 9-s + 4·17-s + 4·19-s − 2·23-s + 25-s + 2·43-s + 2·45-s + 2·59-s − 2·61-s − 4·71-s − 2·79-s + 2·83-s + 8·85-s − 4·89-s + 8·95-s − 2·97-s − 4·103-s + 4·109-s − 4·115-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.606932563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.606932563\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.58868442508291220530219314533, −6.20498344693873524139999229300, −5.83002875426397754698990187127, −5.79070725236204426886621533244, −5.74418082480686659743647899497, −5.71612907731760073144472363523, −5.41811424919950636788680786496, −5.40915711808165841230478498160, −5.15678215541945683702455153714, −4.49378936853491655067481916825, −4.46007901636492515280245027392, −4.42855863092604703206565269037, −4.11407255206273657117370865233, −3.53684830785752280502333142290, −3.44174158920317113721553651336, −3.34184248842188063284839383277, −3.23757710751098721523879279006, −2.70456228909367643321662855237, −2.52090336197369662397391925419, −2.46978555447955656614222619564, −1.83610596099114313247774721518, −1.52338231724967475798673180325, −1.32797191860981369329311876430, −1.14412385779240633446498912733, −1.12215674077858600446124596234,
1.12215674077858600446124596234, 1.14412385779240633446498912733, 1.32797191860981369329311876430, 1.52338231724967475798673180325, 1.83610596099114313247774721518, 2.46978555447955656614222619564, 2.52090336197369662397391925419, 2.70456228909367643321662855237, 3.23757710751098721523879279006, 3.34184248842188063284839383277, 3.44174158920317113721553651336, 3.53684830785752280502333142290, 4.11407255206273657117370865233, 4.42855863092604703206565269037, 4.46007901636492515280245027392, 4.49378936853491655067481916825, 5.15678215541945683702455153714, 5.40915711808165841230478498160, 5.41811424919950636788680786496, 5.71612907731760073144472363523, 5.74418082480686659743647899497, 5.79070725236204426886621533244, 5.83002875426397754698990187127, 6.20498344693873524139999229300, 6.58868442508291220530219314533
