| L(s) = 1 | − 8·11-s + 16·19-s − 4·25-s − 32·41-s − 16·49-s + 32·67-s + 16·73-s − 8·83-s − 8·89-s + 32·97-s + 16·107-s − 72·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 3.67·19-s − 4/5·25-s − 4.99·41-s − 2.28·49-s + 3.90·67-s + 1.87·73-s − 0.878·83-s − 0.847·89-s + 3.24·97-s + 1.54·107-s − 6.77·113-s + 1.81·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5120273647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5120273647\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4$ | \( 1 + 4 T^{2} + 6 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 1254 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 64 T^{2} + 2034 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6918 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 160 T^{2} + 12114 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 2106 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 158 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−5.32218217076338776087378845436, −5.24877125761942395653581980033, −5.07286207799046451139928754220, −4.99544943605955847129179383319, −4.97477997798946062216910752427, −4.53053520632468673635568246616, −4.31595615453133632516911407141, −4.20119693391648844552622810215, −3.77466080009403382984837361321, −3.60067740340767504969527783673, −3.40306428363711878821273363672, −3.39911351030140576005704343636, −3.18165064916747955213383027514, −3.09392918313909945248440701971, −2.62885063979203899564604258674, −2.56868305126763267740821914931, −2.48102673452884035573784223375, −1.96069268115828206256797011800, −1.84098064779980799773282971471, −1.64828622725962021710806935589, −1.49196604579163407520933559571, −1.06271543149966272449908493664, −0.74273844302324181084161323046, −0.54812440877867198459561131451, −0.095885331095539377764696150050,
0.095885331095539377764696150050, 0.54812440877867198459561131451, 0.74273844302324181084161323046, 1.06271543149966272449908493664, 1.49196604579163407520933559571, 1.64828622725962021710806935589, 1.84098064779980799773282971471, 1.96069268115828206256797011800, 2.48102673452884035573784223375, 2.56868305126763267740821914931, 2.62885063979203899564604258674, 3.09392918313909945248440701971, 3.18165064916747955213383027514, 3.39911351030140576005704343636, 3.40306428363711878821273363672, 3.60067740340767504969527783673, 3.77466080009403382984837361321, 4.20119693391648844552622810215, 4.31595615453133632516911407141, 4.53053520632468673635568246616, 4.97477997798946062216910752427, 4.99544943605955847129179383319, 5.07286207799046451139928754220, 5.24877125761942395653581980033, 5.32218217076338776087378845436
