| L(s) = 1 | − 24·17-s − 16·49-s + 24·53-s + 32·61-s − 9·81-s − 16·109-s − 24·113-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 5.82·17-s − 2.28·49-s + 3.29·53-s + 4.09·61-s − 81-s − 1.53·109-s − 2.25·113-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1942549151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1942549151\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.96473337604456935519266085216, −6.93372253683325611998882820066, −6.69401745349335308892127954684, −6.35092770901810737144985724633, −6.08046694615787282544863136460, −6.01938619768155052776068019111, −5.62172684328578717489952923136, −5.37664337051034007770040203822, −5.12365678922605946968864091330, −4.83045740513117562751330629793, −4.71745410487922287316781359951, −4.45400378971733279803856660160, −4.21852707496562364812517222161, −3.93198195935572767804944355242, −3.83138794626566215938153338500, −3.69442272685433421924350737668, −2.92880070839650173753349054373, −2.90428125376114398812313704798, −2.37794954841143723073264539793, −2.34337653681297377569683665906, −2.05388921748437615148930135847, −1.92327126218442408510220523927, −1.28989475626296779572636178527, −0.76161168520116107661676695950, −0.10926304072506061458286929194,
0.10926304072506061458286929194, 0.76161168520116107661676695950, 1.28989475626296779572636178527, 1.92327126218442408510220523927, 2.05388921748437615148930135847, 2.34337653681297377569683665906, 2.37794954841143723073264539793, 2.90428125376114398812313704798, 2.92880070839650173753349054373, 3.69442272685433421924350737668, 3.83138794626566215938153338500, 3.93198195935572767804944355242, 4.21852707496562364812517222161, 4.45400378971733279803856660160, 4.71745410487922287316781359951, 4.83045740513117562751330629793, 5.12365678922605946968864091330, 5.37664337051034007770040203822, 5.62172684328578717489952923136, 6.01938619768155052776068019111, 6.08046694615787282544863136460, 6.35092770901810737144985724633, 6.69401745349335308892127954684, 6.93372253683325611998882820066, 6.96473337604456935519266085216
