| L(s) = 1 | + 4·4-s − 4·7-s + 4·16-s + 2·25-s − 16·28-s − 20·31-s + 18·49-s − 16·64-s − 56·73-s − 20·79-s − 8·97-s + 8·100-s − 56·103-s − 16·112-s − 10·121-s − 80·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1.51·7-s + 16-s + 2/5·25-s − 3.02·28-s − 3.59·31-s + 18/7·49-s − 2·64-s − 6.55·73-s − 2.25·79-s − 0.812·97-s + 4/5·100-s − 5.51·103-s − 1.51·112-s − 0.909·121-s − 7.18·124-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 + 8·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6242118190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6242118190\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2}( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 13 | \( ( 1 - p T^{2} )^{8} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 50 T^{2} + 1659 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 10 T + p T^{2} )^{4}( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + p T^{2} )^{8} \) |
| 43 | \( ( 1 - p T^{2} )^{8} \) |
| 47 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2}( 1 - 94 T^{2} + 6027 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 59 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2}( 1 - 10 T^{2} - 3381 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 61 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 10 T + p T^{2} )^{4}( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 + 134 T^{2} + 11067 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.67948044863502671991038598619, −4.50996030169763667359024946230, −4.45028848747838776523496849396, −4.40297983112379619212009695694, −4.39983156338771042715490629319, −4.05442197590297189624268427393, −3.89972900788846604767602282237, −3.74558361857957389626030641954, −3.58812632601568869054463289983, −3.36127662870721836394116062464, −3.27141121622401510867074093909, −3.16058536683034585584661084168, −2.93362571980390290213624692251, −2.78280534900541210671642549312, −2.74774382084406092953052371020, −2.62407575687882637423001066406, −2.30052497939927283802853469281, −2.14551048995455788175508291742, −1.94638699502647986030422503881, −1.66470680321015291575887856930, −1.57110128565888749918519236109, −1.49133188880425813956927019798, −1.16903776649040967479109837366, −0.57500521998528341156649605637, −0.14016251005132513698023917893,
0.14016251005132513698023917893, 0.57500521998528341156649605637, 1.16903776649040967479109837366, 1.49133188880425813956927019798, 1.57110128565888749918519236109, 1.66470680321015291575887856930, 1.94638699502647986030422503881, 2.14551048995455788175508291742, 2.30052497939927283802853469281, 2.62407575687882637423001066406, 2.74774382084406092953052371020, 2.78280534900541210671642549312, 2.93362571980390290213624692251, 3.16058536683034585584661084168, 3.27141121622401510867074093909, 3.36127662870721836394116062464, 3.58812632601568869054463289983, 3.74558361857957389626030641954, 3.89972900788846604767602282237, 4.05442197590297189624268427393, 4.39983156338771042715490629319, 4.40297983112379619212009695694, 4.45028848747838776523496849396, 4.50996030169763667359024946230, 4.67948044863502671991038598619
Plot not available for L-functions of degree greater than 10.
