| L(s) = 1 | − 7-s − 9-s + 2·11-s + 6·23-s − 2·25-s − 4·29-s − 4·37-s + 4·43-s + 49-s + 8·53-s + 63-s + 16·67-s − 6·71-s − 2·77-s + 16·79-s + 81-s − 2·99-s − 6·107-s − 20·109-s − 4·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1/3·9-s + 0.603·11-s + 1.25·23-s − 2/5·25-s − 0.742·29-s − 0.657·37-s + 0.609·43-s + 1/7·49-s + 1.09·53-s + 0.125·63-s + 1.95·67-s − 0.712·71-s − 0.227·77-s + 1.80·79-s + 1/9·81-s − 0.201·99-s − 0.580·107-s − 1.91·109-s − 0.376·113-s − 0.909·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.634075498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.634075498\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.699282088321589201467643517093, −8.236972050375983718950285796462, −7.77018949902582179915957339205, −7.14070251885670841829509075166, −6.87700863147116782395497369892, −6.39344426492772010571509299168, −5.80637317277794281955672583724, −5.36007472380367461897506998623, −4.92880193215004895162898338699, −4.11584250552478313262890669286, −3.76479384395257081325203691637, −3.13250433380878648175129575563, −2.49956093243554054195671052448, −1.73125840677039039003718737465, −0.72046474937322208138838996608,
0.72046474937322208138838996608, 1.73125840677039039003718737465, 2.49956093243554054195671052448, 3.13250433380878648175129575563, 3.76479384395257081325203691637, 4.11584250552478313262890669286, 4.92880193215004895162898338699, 5.36007472380367461897506998623, 5.80637317277794281955672583724, 6.39344426492772010571509299168, 6.87700863147116782395497369892, 7.14070251885670841829509075166, 7.77018949902582179915957339205, 8.236972050375983718950285796462, 8.699282088321589201467643517093
