| L(s) = 1 | − 4·2-s + 8·4-s + 4·5-s − 8·8-s − 16·10-s − 4·16-s + 32·20-s − 10·25-s + 32·32-s − 32·40-s + 28·41-s + 10·49-s + 40·50-s − 64·64-s − 16·80-s − 18·81-s − 112·82-s + 52·97-s − 40·98-s − 80·100-s − 52·101-s + 20·109-s − 4·113-s + 4·121-s − 80·125-s + 127-s + 64·128-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s + 1.78·5-s − 2.82·8-s − 5.05·10-s − 16-s + 7.15·20-s − 2·25-s + 5.65·32-s − 5.05·40-s + 4.37·41-s + 10/7·49-s + 5.65·50-s − 8·64-s − 1.78·80-s − 2·81-s − 12.3·82-s + 5.27·97-s − 4.04·98-s − 8·100-s − 5.17·101-s + 1.91·109-s − 0.376·113-s + 4/11·121-s − 7.15·125-s + 0.0887·127-s + 5.65·128-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3526701221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3526701221\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 172 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{4} \) |
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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
−9.863151427683455131117468865472, −9.446408232563317604710590508309, −9.353349697608918861458412722843, −9.161264426930919023587310174236, −9.113808475836557078690057824454, −8.499443123739439181716666084497, −8.283268613283376090671820031300, −8.014204942419140880891524487942, −7.61813312078094077886396957695, −7.44872806977170889152838436110, −7.38800892686071741854851483263, −6.65472337270925702087526487951, −6.65246070053453051526810544784, −5.91364629676669535619164427655, −5.83978548326827659751706140955, −5.79093213079397994464081785382, −5.21413755661101449106027053247, −4.52027768891085925261786286561, −4.11195671105746595029527799726, −4.09474349060503262817842733373, −3.01753161726908619384784875676, −2.41862022534931357359943638829, −2.08780428448749403523583696110, −1.78788613401014813332776415773, −0.907116750003131124119676193492,
0.907116750003131124119676193492, 1.78788613401014813332776415773, 2.08780428448749403523583696110, 2.41862022534931357359943638829, 3.01753161726908619384784875676, 4.09474349060503262817842733373, 4.11195671105746595029527799726, 4.52027768891085925261786286561, 5.21413755661101449106027053247, 5.79093213079397994464081785382, 5.83978548326827659751706140955, 5.91364629676669535619164427655, 6.65246070053453051526810544784, 6.65472337270925702087526487951, 7.38800892686071741854851483263, 7.44872806977170889152838436110, 7.61813312078094077886396957695, 8.014204942419140880891524487942, 8.283268613283376090671820031300, 8.499443123739439181716666084497, 9.113808475836557078690057824454, 9.161264426930919023587310174236, 9.353349697608918861458412722843, 9.446408232563317604710590508309, 9.863151427683455131117468865472
