| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 4·11-s − 2·13-s + 4·14-s + 5·16-s − 2·17-s + 6·19-s − 8·22-s + 4·23-s + 4·26-s − 6·28-s + 10·29-s − 6·32-s + 4·34-s + 4·37-s − 12·38-s + 8·41-s + 12·44-s − 8·46-s + 2·47-s + 3·49-s − 6·52-s − 6·53-s + 8·56-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s + 1.20·11-s − 0.554·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s + 1.37·19-s − 1.70·22-s + 0.834·23-s + 0.784·26-s − 1.13·28-s + 1.85·29-s − 1.06·32-s + 0.685·34-s + 0.657·37-s − 1.94·38-s + 1.24·41-s + 1.80·44-s − 1.17·46-s + 0.291·47-s + 3/7·49-s − 0.832·52-s − 0.824·53-s + 1.06·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.580850851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580850851\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 73 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 105 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 161 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 93 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 108 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.83677486316984031957138927481, −7.42659311195923015221784139978, −7.25949067644908833017585668190, −7.07419760363318171650308874754, −6.49777134664145749342429645049, −6.35760217996565339878042730271, −5.93161941873424545532738333880, −5.81599434161416013239979454738, −4.96780600701151585118914247229, −4.93341165670980518877937490351, −4.19781980102160915454060471455, −4.15754538089976478674910026844, −3.31037654377670398631283679486, −3.14239635096686895622078645783, −2.77475296388841735393630008706, −2.42865912675474373957756922350, −1.74956366764449587655182591078, −1.32616081621087052905040583679, −0.867763408722311086404962765673, −0.46756574065012921790133985023,
0.46756574065012921790133985023, 0.867763408722311086404962765673, 1.32616081621087052905040583679, 1.74956366764449587655182591078, 2.42865912675474373957756922350, 2.77475296388841735393630008706, 3.14239635096686895622078645783, 3.31037654377670398631283679486, 4.15754538089976478674910026844, 4.19781980102160915454060471455, 4.93341165670980518877937490351, 4.96780600701151585118914247229, 5.81599434161416013239979454738, 5.93161941873424545532738333880, 6.35760217996565339878042730271, 6.49777134664145749342429645049, 7.07419760363318171650308874754, 7.25949067644908833017585668190, 7.42659311195923015221784139978, 7.83677486316984031957138927481
