| L(s) = 1 | + 8·4-s + 4·13-s + 40·16-s + 24·47-s + 32·52-s + 160·64-s − 16·67-s − 81-s + 20·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + 192·188-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4·4-s + 1.10·13-s + 10·16-s + 3.50·47-s + 4.43·52-s + 20·64-s − 1.95·67-s − 1/9·81-s + 1.97·103-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 14.0·188-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.02886778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.02886778\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 5 | $C_2^3$ | \( 1 - 49 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 73 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 167 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2^3$ | \( 1 + 1442 T^{4} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 626 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 1967 T^{4} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^3$ | \( 1 - 4078 T^{4} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 - 10078 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^3$ | \( 1 - 9118 T^{4} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 14974 T^{4} + p^{4} T^{8} \) |
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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
−7.21066038262213218237400892302, −7.05227474450183067054582144474, −7.00448533804454035030760724083, −6.57043846800013997310569913738, −6.41328134376657296504451112001, −6.02825478777099626855272140848, −6.02496317141060105256383324225, −5.92568082293885500060104968465, −5.67709935703679624950240077231, −5.43742112121643895585287664639, −5.09657817114600806386762546556, −4.63167706717331166262869664550, −4.52533707973565570620088048009, −3.87869288877347300655668065745, −3.74444128469671896603943371645, −3.48893680702854685641541075935, −3.39528092979712272246478327571, −2.83102171316810189339802850338, −2.67955178810694344772057441194, −2.43594688520711333077623419212, −2.28909075395878645021437729485, −1.70001946305155502000231631962, −1.65255687833407967701485660321, −1.06395671732091897442757008800, −0.945693231677351886500393808523,
0.945693231677351886500393808523, 1.06395671732091897442757008800, 1.65255687833407967701485660321, 1.70001946305155502000231631962, 2.28909075395878645021437729485, 2.43594688520711333077623419212, 2.67955178810694344772057441194, 2.83102171316810189339802850338, 3.39528092979712272246478327571, 3.48893680702854685641541075935, 3.74444128469671896603943371645, 3.87869288877347300655668065745, 4.52533707973565570620088048009, 4.63167706717331166262869664550, 5.09657817114600806386762546556, 5.43742112121643895585287664639, 5.67709935703679624950240077231, 5.92568082293885500060104968465, 6.02496317141060105256383324225, 6.02825478777099626855272140848, 6.41328134376657296504451112001, 6.57043846800013997310569913738, 7.00448533804454035030760724083, 7.05227474450183067054582144474, 7.21066038262213218237400892302
