| L(s) = 1 | − 2·5-s − 2·7-s − 4·13-s + 14·19-s + 25-s + 24·29-s + 2·31-s + 4·35-s + 2·37-s − 4·43-s − 12·47-s + 7·49-s − 12·59-s + 8·61-s + 8·65-s + 2·67-s − 10·73-s − 22·79-s + 24·83-s − 12·89-s + 8·91-s − 28·95-s + 32·97-s − 24·101-s + 26·103-s + 2·109-s + 72·113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 1.10·13-s + 3.21·19-s + 1/5·25-s + 4.45·29-s + 0.359·31-s + 0.676·35-s + 0.328·37-s − 0.609·43-s − 1.75·47-s + 49-s − 1.56·59-s + 1.02·61-s + 0.992·65-s + 0.244·67-s − 1.17·73-s − 2.47·79-s + 2.63·83-s − 1.27·89-s + 0.838·91-s − 2.87·95-s + 3.24·97-s − 2.38·101-s + 2.56·103-s + 0.191·109-s + 6.77·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.921057358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921057358\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_2^3$ | \( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 28 T^{2} + 255 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 12 T + 76 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T + 13 T^{2} + 142 T^{3} - 1004 T^{4} + 142 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T - 53 T^{2} + 34 T^{3} + 1732 T^{4} + 34 p T^{5} - 53 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 34 T^{2} - 1653 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 216 p T^{5} + 8 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 2 T^{2} + 448 T^{3} - 3269 T^{4} + 448 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 113 T^{2} + 34 T^{3} + 8932 T^{4} + 34 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 10 T - 53 T^{2} + 70 T^{3} + 10780 T^{4} + 70 p T^{5} - 53 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 184 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T - 52 T^{2} + 216 T^{3} + 17679 T^{4} + 216 p T^{5} - 52 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.95449197521281366842934012007, −6.55257730381509734267040921309, −6.51171990006710673472046886563, −6.31123394377091446421841536434, −6.24575771342782061880304356322, −5.78513153580815635656993120017, −5.65521055207275930482434541682, −5.24305907986460708846884144149, −5.00045797768078099113121897621, −4.82698715394563202024662097146, −4.77369789118127068714612529079, −4.43972873644595143326666936391, −4.43830908923356832439408447243, −3.66265159330290032330311514947, −3.61402089681465346180268872021, −3.45965757342017302028359926700, −3.08693055860762224970762076719, −2.95195106985156711257697974581, −2.72897962206069743905976128045, −2.36515224555647219137289890352, −2.15007980847801511799241417155, −1.32392642536948699133660662833, −1.22003362621220431944909068351, −0.878529131067263648826946604968, −0.35467799964582136721994288193,
0.35467799964582136721994288193, 0.878529131067263648826946604968, 1.22003362621220431944909068351, 1.32392642536948699133660662833, 2.15007980847801511799241417155, 2.36515224555647219137289890352, 2.72897962206069743905976128045, 2.95195106985156711257697974581, 3.08693055860762224970762076719, 3.45965757342017302028359926700, 3.61402089681465346180268872021, 3.66265159330290032330311514947, 4.43830908923356832439408447243, 4.43972873644595143326666936391, 4.77369789118127068714612529079, 4.82698715394563202024662097146, 5.00045797768078099113121897621, 5.24305907986460708846884144149, 5.65521055207275930482434541682, 5.78513153580815635656993120017, 6.24575771342782061880304356322, 6.31123394377091446421841536434, 6.51171990006710673472046886563, 6.55257730381509734267040921309, 6.95449197521281366842934012007
