| L(s) = 1 | − 3-s + 3·4-s + 3·7-s + 9-s − 3·12-s − 13-s + 5·16-s + 11·19-s − 3·21-s + 25-s − 27-s + 9·28-s − 10·31-s + 3·36-s + 11·37-s + 39-s − 20·43-s − 5·48-s + 5·49-s − 3·52-s − 11·57-s − 2·61-s + 3·63-s + 3·64-s + 18·67-s + 10·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 3/2·4-s + 1.13·7-s + 1/3·9-s − 0.866·12-s − 0.277·13-s + 5/4·16-s + 2.52·19-s − 0.654·21-s + 1/5·25-s − 0.192·27-s + 1.70·28-s − 1.79·31-s + 1/2·36-s + 1.80·37-s + 0.160·39-s − 3.04·43-s − 0.721·48-s + 5/7·49-s − 0.416·52-s − 1.45·57-s − 0.256·61-s + 0.377·63-s + 3/8·64-s + 2.19·67-s + 1.17·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174987 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174987 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.453421284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453421284\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 6481 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 96 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−9.361251703219722529818917763072, −8.563653327834395642241942971340, −8.006149911021988209766515355709, −7.62542751655305916764838093940, −7.27125708880942606657757569096, −6.80447826583211080775224732927, −6.32654413357506929961127268815, −5.63295068139686522685632624067, −5.16277703627138315893249231164, −4.97585900084740406545490407960, −3.91814609317369700697614914278, −3.30844602836631101915578491716, −2.59259144229511084339599798371, −1.79752051583702202781856751314, −1.18893358595631416210197796801,
1.18893358595631416210197796801, 1.79752051583702202781856751314, 2.59259144229511084339599798371, 3.30844602836631101915578491716, 3.91814609317369700697614914278, 4.97585900084740406545490407960, 5.16277703627138315893249231164, 5.63295068139686522685632624067, 6.32654413357506929961127268815, 6.80447826583211080775224732927, 7.27125708880942606657757569096, 7.62542751655305916764838093940, 8.006149911021988209766515355709, 8.563653327834395642241942971340, 9.361251703219722529818917763072
