| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 35·16-s + 12·23-s − 20·25-s − 24·29-s + 56·32-s + 48·46-s + 14·49-s − 80·50-s − 96·58-s + 84·64-s − 24·71-s + 120·92-s + 56·98-s − 200·100-s − 240·116-s + 16·121-s + 127-s + 120·128-s + 131-s + 137-s + 139-s − 96·142-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 7.07·8-s + 35/4·16-s + 2.50·23-s − 4·25-s − 4.45·29-s + 9.89·32-s + 7.07·46-s + 2·49-s − 11.3·50-s − 12.6·58-s + 21/2·64-s − 2.84·71-s + 12.5·92-s + 5.65·98-s − 20·100-s − 22.2·116-s + 1.45·121-s + 0.0887·127-s + 10.6·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8.05·142-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.44018529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.44018529\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 166 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.02645522227697458509440499997, −5.97348792941557624661012345547, −5.67308680719579435786922090653, −5.56721320969627415274843338869, −5.53409663702014928442355979111, −5.27571459017992269306066473669, −4.95229236156132962451658598646, −4.77783714742306794607007775278, −4.42588243373610761497166722523, −4.40035835844814674010577515631, −4.14769619263461526932399963377, −3.85887786935502340177450220201, −3.76566088878819622429457664920, −3.55877467082129725539108911657, −3.35821989808786810804865360461, −3.20254442689168276269298587880, −2.78038594758042144963002457247, −2.69630869581882641332146538740, −2.16470466546053627030660265440, −2.04016253243023552101630114160, −2.03559795432283645106603983425, −1.51764621664569877255707739830, −1.46791156819807032432729859684, −0.849280132399421665771530040334, −0.24390512197382370615077495304,
0.24390512197382370615077495304, 0.849280132399421665771530040334, 1.46791156819807032432729859684, 1.51764621664569877255707739830, 2.03559795432283645106603983425, 2.04016253243023552101630114160, 2.16470466546053627030660265440, 2.69630869581882641332146538740, 2.78038594758042144963002457247, 3.20254442689168276269298587880, 3.35821989808786810804865360461, 3.55877467082129725539108911657, 3.76566088878819622429457664920, 3.85887786935502340177450220201, 4.14769619263461526932399963377, 4.40035835844814674010577515631, 4.42588243373610761497166722523, 4.77783714742306794607007775278, 4.95229236156132962451658598646, 5.27571459017992269306066473669, 5.53409663702014928442355979111, 5.56721320969627415274843338869, 5.67308680719579435786922090653, 5.97348792941557624661012345547, 6.02645522227697458509440499997
