L(s) = 1 | + 24·29-s − 8·41-s + 12·49-s − 8·61-s + 40·89-s + 8·101-s + 72·109-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | + 4.45·29-s − 1.24·41-s + 12/7·49-s − 1.02·61-s + 4.23·89-s + 0.796·101-s + 6.89·109-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 + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.406894483\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.406894483\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.61633048900528609860934201220, −5.19218545621568710898378051732, −5.15305494852376585102118142553, −5.01141070697186690976656219957, −4.83531634803333086873164805530, −4.57668138792436507615124268942, −4.42741845741385986054532101570, −4.42686598211880121812867009323, −4.20340810293441425257150978015, −3.70939780375753892015849306796, −3.57835621788388802753566999296, −3.45821674675477819568040917461, −3.35003215205336487942276309473, −2.95501577763324480433586879451, −2.79012952573848362269297956972, −2.69922534532862369397671722693, −2.50122322779165823029852736069, −2.04335727876463045059910511894, −1.87085214321009233565149092071, −1.82703938522790392996229776317, −1.49790607187681915886781130889, −0.879592843478337469356589750622, −0.812763391941497603436675494074, −0.72303286314502468889046315025, −0.41996540591292181751682563827,
0.41996540591292181751682563827, 0.72303286314502468889046315025, 0.812763391941497603436675494074, 0.879592843478337469356589750622, 1.49790607187681915886781130889, 1.82703938522790392996229776317, 1.87085214321009233565149092071, 2.04335727876463045059910511894, 2.50122322779165823029852736069, 2.69922534532862369397671722693, 2.79012952573848362269297956972, 2.95501577763324480433586879451, 3.35003215205336487942276309473, 3.45821674675477819568040917461, 3.57835621788388802753566999296, 3.70939780375753892015849306796, 4.20340810293441425257150978015, 4.42686598211880121812867009323, 4.42741845741385986054532101570, 4.57668138792436507615124268942, 4.83531634803333086873164805530, 5.01141070697186690976656219957, 5.15305494852376585102118142553, 5.19218545621568710898378051732, 5.61633048900528609860934201220