| L(s) = 1 | − 4·9-s − 16·17-s + 8·25-s − 48·41-s − 56·49-s − 16·73-s + 10·81-s − 48·89-s − 48·97-s + 48·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 64·153-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 3.88·17-s + 8/5·25-s − 7.49·41-s − 8·49-s − 1.87·73-s + 10/9·81-s − 5.08·89-s − 4.87·97-s + 4.51·113-s − 0.363·121-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 + 5.17·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05843275535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05843275535\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| 11 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 + p T^{2} )^{8} \) |
| 13 | \( ( 1 - 12 T^{2} - 10 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 52 T^{2} + 1350 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 76 T^{2} + 2742 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 36 T^{2} + 710 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 148 T^{2} + 9510 T^{4} + 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 100 T^{2} + 6582 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 12 T^{2} + 854 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 76 T^{2} + 5430 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 44 T^{2} + 3318 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 116 T^{2} + 9990 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.80876896965242566021009128175, −3.56137136424758442918067431394, −3.53343119841209111722155295700, −3.41808033391836452614725319320, −3.26512073079775445908732706493, −3.12820948660876013456515855688, −3.08138725606455416524843589333, −3.07793717713777225861240181179, −2.75784816381058023171367726900, −2.64581290731530946762623453146, −2.62347172807008920492244998320, −2.62258071977936047522231867837, −2.33120245418468535845523492851, −2.02798369308627358589081783558, −1.84477891261726450563088471071, −1.84194544976763737946892223749, −1.66758245912943669175196845977, −1.53147994610171845358541093915, −1.51374162101673647367645369940, −1.41398146755603889517428755138, −1.28879741259057843093242668681, −0.56451533337255460093573152951, −0.43920816540988935868016217941, −0.26317831998377196098801265789, −0.05573700954874060384421435263,
0.05573700954874060384421435263, 0.26317831998377196098801265789, 0.43920816540988935868016217941, 0.56451533337255460093573152951, 1.28879741259057843093242668681, 1.41398146755603889517428755138, 1.51374162101673647367645369940, 1.53147994610171845358541093915, 1.66758245912943669175196845977, 1.84194544976763737946892223749, 1.84477891261726450563088471071, 2.02798369308627358589081783558, 2.33120245418468535845523492851, 2.62258071977936047522231867837, 2.62347172807008920492244998320, 2.64581290731530946762623453146, 2.75784816381058023171367726900, 3.07793717713777225861240181179, 3.08138725606455416524843589333, 3.12820948660876013456515855688, 3.26512073079775445908732706493, 3.41808033391836452614725319320, 3.53343119841209111722155295700, 3.56137136424758442918067431394, 3.80876896965242566021009128175
Plot not available for L-functions of degree greater than 10.
