| L(s) = 1 | + 4-s − 2·7-s − 8·13-s + 16-s + 4·19-s − 2·28-s − 4·31-s − 20·37-s + 10·43-s − 9·49-s − 8·52-s + 14·61-s + 5·64-s + 16·67-s − 14·73-s + 4·76-s − 16·79-s + 16·91-s + 16·97-s + 52·103-s + 44·109-s − 2·112-s − 17·121-s − 4·124-s + 127-s + 131-s − 8·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 2.21·13-s + 1/4·16-s + 0.917·19-s − 0.377·28-s − 0.718·31-s − 3.28·37-s + 1.52·43-s − 9/7·49-s − 1.10·52-s + 1.79·61-s + 5/8·64-s + 1.95·67-s − 1.63·73-s + 0.458·76-s − 1.80·79-s + 1.67·91-s + 1.62·97-s + 5.12·103-s + 4.21·109-s − 0.188·112-s − 1.54·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.907003577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.907003577\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 17 T^{2} + 240 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 53 T^{2} + 1272 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 44 T^{2} + 1014 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 68 T^{2} + 2310 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 128 T^{2} + 7326 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 89 T^{2} + 5400 T^{4} + 89 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 28 T^{2} + 5286 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 92 T^{2} + 6966 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 + 92 T^{2} + 3750 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 188 T^{2} + 17862 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 116 T^{2} + 18678 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.95246750070678135053446123399, −5.62565567288265603421221408236, −5.49837055335831154596819657063, −5.30414609316886304049183975938, −5.28177288860000187406576089339, −4.86258779772334871147392702510, −4.83948118405650059980035898407, −4.54599532496041508820666990428, −4.48144268941522554118218982024, −4.10954196088245437901309916767, −3.78838722298482149956838015853, −3.63077822889216765878315600053, −3.40190423354009054247757610037, −3.29674291406625867574955778236, −3.09648093233906247320946157074, −2.88205191504395570446803560804, −2.46468861792370073945253136611, −2.27132020531464096992493905499, −2.23859557598730157798145403507, −1.80809905665667816975886975958, −1.69549212233742113033041263377, −1.40655745527787418380906670269, −0.67402213270281465886522456170, −0.66338135046827987866037205820, −0.30773948972674420177651002896,
0.30773948972674420177651002896, 0.66338135046827987866037205820, 0.67402213270281465886522456170, 1.40655745527787418380906670269, 1.69549212233742113033041263377, 1.80809905665667816975886975958, 2.23859557598730157798145403507, 2.27132020531464096992493905499, 2.46468861792370073945253136611, 2.88205191504395570446803560804, 3.09648093233906247320946157074, 3.29674291406625867574955778236, 3.40190423354009054247757610037, 3.63077822889216765878315600053, 3.78838722298482149956838015853, 4.10954196088245437901309916767, 4.48144268941522554118218982024, 4.54599532496041508820666990428, 4.83948118405650059980035898407, 4.86258779772334871147392702510, 5.28177288860000187406576089339, 5.30414609316886304049183975938, 5.49837055335831154596819657063, 5.62565567288265603421221408236, 5.95246750070678135053446123399
