| L(s) = 1 | + 8·3-s + 32·9-s − 32·16-s + 88·19-s + 48·25-s + 72·27-s − 52·31-s + 68·43-s − 256·48-s + 146·49-s + 704·57-s + 268·61-s − 400·67-s − 304·73-s + 384·75-s + 47·81-s − 416·93-s − 100·97-s + 4·103-s + 396·109-s + 127-s + 544·129-s + 131-s + 137-s + 139-s − 1.02e3·144-s + 1.16e3·147-s + ⋯ |
| L(s) = 1 | + 8/3·3-s + 32/9·9-s − 2·16-s + 4.63·19-s + 1.91·25-s + 8/3·27-s − 1.67·31-s + 1.58·43-s − 5.33·48-s + 2.97·49-s + 12.3·57-s + 4.39·61-s − 5.97·67-s − 4.16·73-s + 5.11·75-s + 0.580·81-s − 4.47·93-s − 1.03·97-s + 0.0388·103-s + 3.63·109-s + 0.00787·127-s + 4.21·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.11·144-s + 7.94·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.347535282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.347535282\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 73 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 23953 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 119953 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 44 T + 968 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 178993 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 224 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 26 T + 338 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2513 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 1084753 T^{4} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 34 T + 578 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 784 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 13910639 T^{4} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 18768478 T^{4} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 67 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + 100 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 + 2073167 T^{4} + p^{8} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 4561 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 5066 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 123934367 T^{4} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.983778033987441202315107925259, −8.905905935234667331588780067270, −8.673767858929156923404660997975, −7.972136456950412413237786555144, −7.75899826693347072243660340303, −7.70252407730134744849125046187, −7.31291958751473368567482752119, −7.06853647537111344488803599267, −7.02380470192875419509232447698, −6.87422290070678579695976874908, −5.99125190313169500402224879531, −5.70342643038456322478247373993, −5.42109723468143709036723745312, −5.38504310207685658370972249340, −4.63379491403698956166341390953, −4.37443223710204944055053514711, −4.23103267147457906194518557479, −3.50122148280273341279165607823, −3.48536693513666846467911814275, −2.87654534543550957148819516136, −2.80829110743917917383870220929, −2.58687963085365855790192409372, −1.88834231531864293736918129943, −1.38502037939227005404942432402, −0.809907518020263380382782881715,
0.809907518020263380382782881715, 1.38502037939227005404942432402, 1.88834231531864293736918129943, 2.58687963085365855790192409372, 2.80829110743917917383870220929, 2.87654534543550957148819516136, 3.48536693513666846467911814275, 3.50122148280273341279165607823, 4.23103267147457906194518557479, 4.37443223710204944055053514711, 4.63379491403698956166341390953, 5.38504310207685658370972249340, 5.42109723468143709036723745312, 5.70342643038456322478247373993, 5.99125190313169500402224879531, 6.87422290070678579695976874908, 7.02380470192875419509232447698, 7.06853647537111344488803599267, 7.31291958751473368567482752119, 7.70252407730134744849125046187, 7.75899826693347072243660340303, 7.972136456950412413237786555144, 8.673767858929156923404660997975, 8.905905935234667331588780067270, 8.983778033987441202315107925259
