| L(s) = 1 | + 4·3-s + 4·9-s − 4·27-s + 8·31-s − 8·37-s − 8·41-s − 28·43-s + 20·49-s − 32·53-s − 36·67-s − 8·71-s − 32·79-s − 10·81-s + 12·83-s − 8·89-s + 32·93-s − 4·107-s − 32·111-s + 36·121-s − 32·123-s + 127-s − 112·129-s + 131-s + 137-s + 139-s + 80·147-s + 149-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 4/3·9-s − 0.769·27-s + 1.43·31-s − 1.31·37-s − 1.24·41-s − 4.26·43-s + 20/7·49-s − 4.39·53-s − 4.39·67-s − 0.949·71-s − 3.60·79-s − 1.11·81-s + 1.31·83-s − 0.847·89-s + 3.31·93-s − 0.386·107-s − 3.03·111-s + 3.27·121-s − 2.88·123-s + 0.0887·127-s − 9.86·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.59·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.218964608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218964608\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1274 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.30400507093415134794035337227, −7.11557771179290550011259327190, −7.10814942907766504385392273905, −6.93117411464043342299825951590, −6.35806061421585888655390720022, −6.24051649265332756439272833205, −5.98749488844225468402154746379, −5.97053639946335435765034744529, −5.40453170544000197434787236152, −5.23137781273671308294036010667, −4.88407789672504954709828188624, −4.65616410189221536010085288205, −4.49762658742986309713201419060, −4.29686576740692511376344309883, −3.63341304767975850645846690948, −3.58497330970638344912059166494, −3.19875189008048115435963113536, −3.18200286231576992832809567057, −2.78690253514986451073744104943, −2.75731264472808744127891687876, −2.34239977120619844696625547455, −1.64427633098665513122087339943, −1.64377486087701292361627624841, −1.44042933670270475032837919096, −0.20530186332446194564206081423,
0.20530186332446194564206081423, 1.44042933670270475032837919096, 1.64377486087701292361627624841, 1.64427633098665513122087339943, 2.34239977120619844696625547455, 2.75731264472808744127891687876, 2.78690253514986451073744104943, 3.18200286231576992832809567057, 3.19875189008048115435963113536, 3.58497330970638344912059166494, 3.63341304767975850645846690948, 4.29686576740692511376344309883, 4.49762658742986309713201419060, 4.65616410189221536010085288205, 4.88407789672504954709828188624, 5.23137781273671308294036010667, 5.40453170544000197434787236152, 5.97053639946335435765034744529, 5.98749488844225468402154746379, 6.24051649265332756439272833205, 6.35806061421585888655390720022, 6.93117411464043342299825951590, 7.10814942907766504385392273905, 7.11557771179290550011259327190, 7.30400507093415134794035337227
