| L(s) = 1 | + 4-s − 25-s − 6·31-s − 64-s − 2·79-s − 100-s − 121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 4-s − 25-s − 6·31-s − 64-s − 2·79-s − 100-s − 121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2014906873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2014906873\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 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
−6.22331809369899560425638436067, −6.01678690582957789126432851271, −5.75898325855555291322067703194, −5.73354281641219609440268816672, −5.48225692302743293197348066355, −5.40325792910807685579913033669, −5.03751937344618109071226814587, −4.98631094857944052906232788493, −4.71040650775833899404681191272, −4.31656233748741455226282786416, −4.19618467312770418752699434175, −3.85595229068464594085815814716, −3.84982242891029496866036603075, −3.52796638331368601987844280031, −3.36216327225643675048503068124, −3.28503536118662693449654919080, −2.80345752791468587936824401929, −2.49854503131194138573829266227, −2.29776339363999742878148785697, −2.27172396268634410937145333381, −1.81512168043010021678651177499, −1.65469086099711228253362739435, −1.30068320453542783647605866453, −1.26662938912428378822387550457, −0.14187585088491212629746735570,
0.14187585088491212629746735570, 1.26662938912428378822387550457, 1.30068320453542783647605866453, 1.65469086099711228253362739435, 1.81512168043010021678651177499, 2.27172396268634410937145333381, 2.29776339363999742878148785697, 2.49854503131194138573829266227, 2.80345752791468587936824401929, 3.28503536118662693449654919080, 3.36216327225643675048503068124, 3.52796638331368601987844280031, 3.84982242891029496866036603075, 3.85595229068464594085815814716, 4.19618467312770418752699434175, 4.31656233748741455226282786416, 4.71040650775833899404681191272, 4.98631094857944052906232788493, 5.03751937344618109071226814587, 5.40325792910807685579913033669, 5.48225692302743293197348066355, 5.73354281641219609440268816672, 5.75898325855555291322067703194, 6.01678690582957789126432851271, 6.22331809369899560425638436067
