| L(s) = 1 | − 2·3-s + 4·9-s + 14·13-s − 6·17-s + 12·19-s + 12·23-s − 2·25-s − 4·27-s + 12·29-s + 18·37-s − 28·39-s − 36·41-s + 4·43-s − 5·49-s + 12·51-s − 12·53-s − 24·57-s − 36·59-s − 2·61-s − 24·69-s + 4·75-s − 4·79-s + 5·81-s − 24·87-s − 18·89-s + 42·97-s − 18·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 4/3·9-s + 3.88·13-s − 1.45·17-s + 2.75·19-s + 2.50·23-s − 2/5·25-s − 0.769·27-s + 2.22·29-s + 2.95·37-s − 4.48·39-s − 5.62·41-s + 0.609·43-s − 5/7·49-s + 1.68·51-s − 1.64·53-s − 3.17·57-s − 4.68·59-s − 0.256·61-s − 2.88·69-s + 0.461·75-s − 0.450·79-s + 5/9·81-s − 2.57·87-s − 1.90·89-s + 4.26·97-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.447597346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.447597346\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - 4 T^{3} - 5 T^{4} - 4 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 20 T^{2} - 108 T^{3} - 645 T^{4} - 108 p T^{5} + 20 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 89 T^{2} - 492 T^{3} + 2232 T^{4} - 492 p T^{5} + 89 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 74 T^{2} - 288 T^{3} + 1059 T^{4} - 288 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 12 T + 62 T^{2} - 288 T^{3} + 1707 T^{4} - 288 p T^{5} + 62 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 18 T + 173 T^{2} - 1170 T^{3} + 6852 T^{4} - 1170 p T^{5} + 173 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 18 T + 149 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 92 T^{2} - 52 T^{3} + 5251 T^{4} - 52 p T^{5} - 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11802 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 132 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 29706 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 5412 T^{4} + 1098 p T^{5} + 169 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 42 T + 920 T^{2} - 13944 T^{3} + 157851 T^{4} - 13944 p T^{5} + 920 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.11041791473494468128114133687, −6.68691676181667507311216926048, −6.52253756228182231270517041633, −6.38453191167161636912746174753, −6.27950395794813015520817028944, −6.19179922846839606642764548718, −5.72229010173481100052549280894, −5.62337398653843095869496329770, −5.33549530626607190735169286313, −4.83086463081268090677316218374, −4.82128859564282290914470308450, −4.65470650329626435820601175594, −4.52961724630950681681669507686, −4.20556699971058825703679206421, −3.67216534060348878440151435883, −3.32783945284583886057813264868, −3.27289180049381840529714993327, −3.10560466506255441381668625812, −3.06321500497633748730980651939, −2.32364198573927201060131854062, −1.66400554434965065468869114701, −1.55902382002382094108792899104, −1.17443018344629294372952734912, −1.09723066898330222835607086177, −0.52893395455188161064031907889,
0.52893395455188161064031907889, 1.09723066898330222835607086177, 1.17443018344629294372952734912, 1.55902382002382094108792899104, 1.66400554434965065468869114701, 2.32364198573927201060131854062, 3.06321500497633748730980651939, 3.10560466506255441381668625812, 3.27289180049381840529714993327, 3.32783945284583886057813264868, 3.67216534060348878440151435883, 4.20556699971058825703679206421, 4.52961724630950681681669507686, 4.65470650329626435820601175594, 4.82128859564282290914470308450, 4.83086463081268090677316218374, 5.33549530626607190735169286313, 5.62337398653843095869496329770, 5.72229010173481100052549280894, 6.19179922846839606642764548718, 6.27950395794813015520817028944, 6.38453191167161636912746174753, 6.52253756228182231270517041633, 6.68691676181667507311216926048, 7.11041791473494468128114133687
