| L(s) = 1 | − 8·5-s + 12·13-s + 24·17-s + 32·25-s − 16·29-s + 12·37-s − 8·49-s − 16·53-s + 12·61-s − 96·65-s − 192·85-s − 16·97-s − 32·101-s − 28·109-s − 24·113-s − 104·125-s + 127-s + 131-s + 137-s + 139-s + 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 3.32·13-s + 5.82·17-s + 32/5·25-s − 2.97·29-s + 1.97·37-s − 8/7·49-s − 2.19·53-s + 1.53·61-s − 11.9·65-s − 20.8·85-s − 1.62·97-s − 3.18·101-s − 2.68·109-s − 2.25·113-s − 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.870932961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.870932961\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
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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
−6.46787669523686087911171125593, −6.12037577768347044742358463764, −5.96017808759052343894584113349, −5.67748616431684756353462742180, −5.48051986940857899689547142481, −5.46575774805932600455139064870, −5.16784833704585551655689096009, −5.11198517483063446117853502887, −4.53037231747636614780756066953, −4.19038354001089997481501250416, −4.10765781821088163764500017615, −3.97848449908377327645617499458, −3.76489249449833151399478317322, −3.62421376643498998759795845427, −3.44187378014585571285971055976, −3.25648033433620113852056975679, −3.14274292787743549056078545070, −2.73742679783199083848775227706, −2.66509754649516254128699670644, −1.73884290339946242779963227575, −1.36442122673214005782188696933, −1.33475434791271000162803324306, −1.20732504073449967939306802557, −0.68095069863424648489079581465, −0.31732378595242006805976043831,
0.31732378595242006805976043831, 0.68095069863424648489079581465, 1.20732504073449967939306802557, 1.33475434791271000162803324306, 1.36442122673214005782188696933, 1.73884290339946242779963227575, 2.66509754649516254128699670644, 2.73742679783199083848775227706, 3.14274292787743549056078545070, 3.25648033433620113852056975679, 3.44187378014585571285971055976, 3.62421376643498998759795845427, 3.76489249449833151399478317322, 3.97848449908377327645617499458, 4.10765781821088163764500017615, 4.19038354001089997481501250416, 4.53037231747636614780756066953, 5.11198517483063446117853502887, 5.16784833704585551655689096009, 5.46575774805932600455139064870, 5.48051986940857899689547142481, 5.67748616431684756353462742180, 5.96017808759052343894584113349, 6.12037577768347044742358463764, 6.46787669523686087911171125593
