| L(s) = 1 | + 2·2-s + 4-s − 2·7-s − 2·8-s + 6·11-s + 2·13-s − 4·14-s − 4·16-s − 12·17-s − 4·19-s + 12·22-s + 6·23-s + 7·25-s + 4·26-s − 2·28-s + 6·29-s + 2·31-s − 2·32-s − 24·34-s + 8·37-s − 8·38-s + 12·41-s − 4·43-s + 6·44-s + 12·46-s − 6·47-s + 49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.755·7-s − 0.707·8-s + 1.80·11-s + 0.554·13-s − 1.06·14-s − 16-s − 2.91·17-s − 0.917·19-s + 2.55·22-s + 1.25·23-s + 7/5·25-s + 0.784·26-s − 0.377·28-s + 1.11·29-s + 0.359·31-s − 0.353·32-s − 4.11·34-s + 1.31·37-s − 1.29·38-s + 1.87·41-s − 0.609·43-s + 0.904·44-s + 1.76·46-s − 0.875·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.438175530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.438175530\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 8 T^{2} - 36 T^{3} + 267 T^{4} - 36 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 16 T^{2} - 36 T^{3} + 1347 T^{4} - 36 p T^{5} - 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T - 19 T^{2} + 18 T^{3} + 1140 T^{4} + 18 p T^{5} - 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T - 32 T^{2} + 52 T^{3} + 211 T^{4} + 52 p T^{5} - 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 38 T^{2} - 288 T^{3} + 3651 T^{4} - 288 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T + 34 T^{2} - 416 T^{3} - 2213 T^{4} - 416 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T - 40 T^{2} - 108 T^{3} + 1875 T^{4} - 108 p T^{5} - 40 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 108 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2 T - 11 T^{2} + 214 T^{3} - 3740 T^{4} + 214 p T^{5} - 11 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 104 T^{2} + 52 T^{3} + 6907 T^{4} + 52 p T^{5} - 104 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 135 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 2 T - 128 T^{2} + 52 T^{3} + 10867 T^{4} + 52 p T^{5} - 128 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 136 T^{2} - 36 T^{3} + 19707 T^{4} - 36 p T^{5} - 136 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 247 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.77823557383799948722601038240, −6.46583651900422204461457243042, −6.45473547018816935600626349402, −6.41733918863507800066994846354, −6.28352584056085649683846085658, −6.13583817042647730762954402209, −5.58294062194433010786551089133, −5.56015334720838172277636743978, −4.97344963616542716136697732867, −4.77043539528773739842918602460, −4.71519917583657309817246387580, −4.51264745035404419538481154228, −4.25634080333921945733477776879, −4.19328386521004793904242747645, −3.88585515206760635598243011174, −3.35374773134845549697044851615, −3.31126729417082316705035204194, −3.07627098327108403018735489422, −2.66286898990005208061405875466, −2.64222264548782763048959330132, −2.01298696169479382691030566272, −1.85237491631129760380828367084, −1.27311627212400670828822401088, −0.978629771851916526419799553484, −0.33386906735975994085501704648,
0.33386906735975994085501704648, 0.978629771851916526419799553484, 1.27311627212400670828822401088, 1.85237491631129760380828367084, 2.01298696169479382691030566272, 2.64222264548782763048959330132, 2.66286898990005208061405875466, 3.07627098327108403018735489422, 3.31126729417082316705035204194, 3.35374773134845549697044851615, 3.88585515206760635598243011174, 4.19328386521004793904242747645, 4.25634080333921945733477776879, 4.51264745035404419538481154228, 4.71519917583657309817246387580, 4.77043539528773739842918602460, 4.97344963616542716136697732867, 5.56015334720838172277636743978, 5.58294062194433010786551089133, 6.13583817042647730762954402209, 6.28352584056085649683846085658, 6.41733918863507800066994846354, 6.45473547018816935600626349402, 6.46583651900422204461457243042, 6.77823557383799948722601038240
