| L(s) = 1 | + 4·19-s + 24·29-s − 4·31-s − 24·41-s + 20·49-s + 24·59-s − 16·61-s − 24·71-s + 16·79-s − 24·101-s + 4·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 0.917·19-s + 4.45·29-s − 0.718·31-s − 3.74·41-s + 20/7·49-s + 3.12·59-s − 2.04·61-s − 2.84·71-s + 1.80·79-s − 2.38·101-s + 0.383·109-s − 3.45·121-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 + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.904025527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904025527\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 954 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4746 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6762 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 5130 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 2166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 16794 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 8586 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 380 T^{2} + 54906 T^{4} - 380 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.40528842672002556285390491265, −5.34200831167183200775491970569, −5.17553230044184579934163596062, −4.85672726409979976307751495546, −4.83428259123963801455432860241, −4.62954806244140513454880081954, −4.46196885201826263911358879286, −4.03687227080622406513105036695, −4.02307549549852819407144278773, −3.83911474139558682258485030435, −3.71411330143282053361577324111, −3.21989197836364247643755961681, −3.19946138318874930605478935298, −2.97206610722112583193702064000, −2.88771833616276332180968625895, −2.59190134347928429341286424265, −2.36772493154904317298346078560, −2.15401280280324063304424457901, −1.96559437160426288413765850720, −1.54321194333131325202879722906, −1.28545367012722313446893825016, −1.11737561732758244511718546820, −1.00600560768580006345935959345, −0.56443098735172459396247610140, −0.16648636980915045512913289406,
0.16648636980915045512913289406, 0.56443098735172459396247610140, 1.00600560768580006345935959345, 1.11737561732758244511718546820, 1.28545367012722313446893825016, 1.54321194333131325202879722906, 1.96559437160426288413765850720, 2.15401280280324063304424457901, 2.36772493154904317298346078560, 2.59190134347928429341286424265, 2.88771833616276332180968625895, 2.97206610722112583193702064000, 3.19946138318874930605478935298, 3.21989197836364247643755961681, 3.71411330143282053361577324111, 3.83911474139558682258485030435, 4.02307549549852819407144278773, 4.03687227080622406513105036695, 4.46196885201826263911358879286, 4.62954806244140513454880081954, 4.83428259123963801455432860241, 4.85672726409979976307751495546, 5.17553230044184579934163596062, 5.34200831167183200775491970569, 5.40528842672002556285390491265
