| L(s) = 1 | − 2·4-s + 3·16-s − 12·23-s + 14·25-s + 12·29-s − 4·43-s + 4·49-s − 12·53-s + 40·61-s − 4·64-s − 8·79-s + 24·92-s − 28·100-s + 36·101-s − 4·103-s − 12·107-s − 24·113-s − 24·116-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4-s + 3/4·16-s − 2.50·23-s + 14/5·25-s + 2.22·29-s − 0.609·43-s + 4/7·49-s − 1.64·53-s + 5.12·61-s − 1/2·64-s − 0.900·79-s + 2.50·92-s − 2.79·100-s + 3.58·101-s − 0.394·103-s − 1.16·107-s − 2.25·113-s − 2.22·116-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.481349028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481349028\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 195 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 20346 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.23219618402024318917055737516, −5.94968681617951073683584634279, −5.71791356953175737763660379785, −5.61531728575718966664116971954, −5.15967365969020026069904649934, −5.10717238815327686419250091804, −5.07554022201380351734413285253, −4.66487284234912939850531083849, −4.60839319173876106184715581880, −4.38386767823403846317582874454, −4.11283763399458967427802181847, −3.82078439526043869015774741772, −3.76992010708974854253397926124, −3.46794654278528226964588061284, −3.31088023549051013013276678955, −2.88026134974295139236782063997, −2.64823604294301214148879193302, −2.61812365475114739270998892494, −2.27927250007156010545737945339, −1.88215650738788830666285664808, −1.66520142302483777679319300576, −1.26387557611753727194401765180, −0.855595296510717856636467497669, −0.76342137354852615322798371255, −0.30300188091268477826440440497,
0.30300188091268477826440440497, 0.76342137354852615322798371255, 0.855595296510717856636467497669, 1.26387557611753727194401765180, 1.66520142302483777679319300576, 1.88215650738788830666285664808, 2.27927250007156010545737945339, 2.61812365475114739270998892494, 2.64823604294301214148879193302, 2.88026134974295139236782063997, 3.31088023549051013013276678955, 3.46794654278528226964588061284, 3.76992010708974854253397926124, 3.82078439526043869015774741772, 4.11283763399458967427802181847, 4.38386767823403846317582874454, 4.60839319173876106184715581880, 4.66487284234912939850531083849, 5.07554022201380351734413285253, 5.10717238815327686419250091804, 5.15967365969020026069904649934, 5.61531728575718966664116971954, 5.71791356953175737763660379785, 5.94968681617951073683584634279, 6.23219618402024318917055737516
