| L(s) = 1 | − 4-s − 6·13-s − 3·16-s − 5·19-s + 7·25-s + 10·31-s − 15·37-s − 5·43-s − 13·49-s + 6·52-s − 4·61-s + 7·64-s − 11·67-s − 19·73-s + 5·76-s − 11·79-s + 20·97-s − 7·100-s + 103-s + 19·109-s + 2·121-s − 10·124-s + 127-s + 131-s + 137-s + 139-s + 15·148-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.66·13-s − 3/4·16-s − 1.14·19-s + 7/5·25-s + 1.79·31-s − 2.46·37-s − 0.762·43-s − 1.85·49-s + 0.832·52-s − 0.512·61-s + 7/8·64-s − 1.34·67-s − 2.22·73-s + 0.573·76-s − 1.23·79-s + 2.03·97-s − 0.699·100-s + 0.0985·103-s + 1.81·109-s + 2/11·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.23·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60021 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60021 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.853147714430869559661654481631, −9.054470991055929134560984744901, −8.697088246244475345360338132408, −8.408292117839778553074169669033, −7.56614212080864805189360801759, −7.10607228411788864459348125307, −6.61938534045764569860186014142, −6.08900615718961029359574417717, −5.13258689173576322757449888775, −4.68084619582272881937449633156, −4.49150326378162214789882353151, −3.33211726645064170222956810602, −2.70560719836692752075649657244, −1.77995467165118306253850006203, 0,
1.77995467165118306253850006203, 2.70560719836692752075649657244, 3.33211726645064170222956810602, 4.49150326378162214789882353151, 4.68084619582272881937449633156, 5.13258689173576322757449888775, 6.08900615718961029359574417717, 6.61938534045764569860186014142, 7.10607228411788864459348125307, 7.56614212080864805189360801759, 8.408292117839778553074169669033, 8.697088246244475345360338132408, 9.054470991055929134560984744901, 9.853147714430869559661654481631
