| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 9-s − 2·11-s − 16-s − 2·17-s − 18-s − 19-s + 2·22-s + 24-s − 25-s + 2·27-s − 2·33-s + 2·34-s + 38-s + 41-s − 2·43-s − 48-s + 2·49-s + 50-s − 2·51-s − 2·54-s − 57-s + 59-s + 64-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 9-s − 2·11-s − 16-s − 2·17-s − 18-s − 19-s + 2·22-s + 24-s − 25-s + 2·27-s − 2·33-s + 2·34-s + 38-s + 41-s − 2·43-s − 48-s + 2·49-s + 50-s − 2·51-s − 2·54-s − 57-s + 59-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3076237393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3076237393\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−13.25814109219072467176254385597, −13.15459683331786685532889254614, −12.81533261395288879606324937369, −12.02361452398343865595862925576, −10.98704515428475493520638789862, −10.98317146030726192002502408597, −10.12876269430643243027513528919, −10.08373028466783430772051515063, −9.327406285338906020916266976238, −8.657723995035487315844952114077, −8.410114434551979520735328937624, −8.101073976834170344302537445441, −7.18947349529932496515057948897, −7.05485255925857981371645894753, −6.06133272168678729103772022551, −5.05189013214110699027880545501, −4.54406087258176959610073463060, −3.85397393163801438416186496316, −2.56999500303935729279473701049, −2.12233152739616025475607167415,
2.12233152739616025475607167415, 2.56999500303935729279473701049, 3.85397393163801438416186496316, 4.54406087258176959610073463060, 5.05189013214110699027880545501, 6.06133272168678729103772022551, 7.05485255925857981371645894753, 7.18947349529932496515057948897, 8.101073976834170344302537445441, 8.410114434551979520735328937624, 8.657723995035487315844952114077, 9.327406285338906020916266976238, 10.08373028466783430772051515063, 10.12876269430643243027513528919, 10.98317146030726192002502408597, 10.98704515428475493520638789862, 12.02361452398343865595862925576, 12.81533261395288879606324937369, 13.15459683331786685532889254614, 13.25814109219072467176254385597
