| L(s) = 1 | − 2·4-s − 12·9-s + 16·11-s − 5·16-s − 8·19-s + 8·29-s − 16·31-s + 24·36-s + 4·41-s − 32·44-s − 36·61-s + 20·64-s + 16·76-s − 8·79-s + 90·81-s − 28·89-s − 192·99-s − 4·101-s − 40·109-s − 16·116-s + 128·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s + 60·144-s + ⋯ |
| L(s) = 1 | − 4-s − 4·9-s + 4.82·11-s − 5/4·16-s − 1.83·19-s + 1.48·29-s − 2.87·31-s + 4·36-s + 0.624·41-s − 4.82·44-s − 4.60·61-s + 5/2·64-s + 1.83·76-s − 0.900·79-s + 10·81-s − 2.96·89-s − 19.2·99-s − 0.398·101-s − 3.83·109-s − 1.48·116-s + 11.6·121-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2662211314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2662211314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 142 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 238 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 2702 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 6398 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 174 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.47765071589904209981988582375, −6.99458235093351901978975708257, −6.85405300086161632701155170251, −6.77370353448409847282737049181, −6.61777229723058957487742732744, −6.08766534998906983357659954491, −6.05898299075652317881794078601, −6.03635344056750740421106512846, −5.68462271901943191206588834853, −5.47590273613522818991366041855, −5.06820405621799394898790781356, −4.75549265831160926968764724472, −4.35750939959986218037553529475, −4.27906155687111406037939896444, −4.18362870055294490420144220124, −3.78219528517658491590116549786, −3.52895563092999420150973342314, −3.20244937692115947667111500139, −2.88885835173010484546137915022, −2.73983846242147603543495171411, −2.13988931774515426549444102533, −1.69515584243645869724782583054, −1.65878360496941161617950942535, −0.837137830147755927829404682578, −0.17321245809921928363541193332,
0.17321245809921928363541193332, 0.837137830147755927829404682578, 1.65878360496941161617950942535, 1.69515584243645869724782583054, 2.13988931774515426549444102533, 2.73983846242147603543495171411, 2.88885835173010484546137915022, 3.20244937692115947667111500139, 3.52895563092999420150973342314, 3.78219528517658491590116549786, 4.18362870055294490420144220124, 4.27906155687111406037939896444, 4.35750939959986218037553529475, 4.75549265831160926968764724472, 5.06820405621799394898790781356, 5.47590273613522818991366041855, 5.68462271901943191206588834853, 6.03635344056750740421106512846, 6.05898299075652317881794078601, 6.08766534998906983357659954491, 6.61777229723058957487742732744, 6.77370353448409847282737049181, 6.85405300086161632701155170251, 6.99458235093351901978975708257, 7.47765071589904209981988582375
