| L(s) = 1 | − 2·3-s + 3·9-s − 2·11-s + 2·17-s + 14·19-s − 25-s − 4·27-s + 4·33-s + 14·41-s + 2·43-s + 2·49-s − 4·51-s − 28·57-s + 20·59-s + 24·67-s − 28·73-s + 2·75-s + 5·81-s + 28·83-s + 16·89-s + 24·97-s − 6·99-s − 26·107-s − 38·113-s − 19·121-s − 28·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 0.603·11-s + 0.485·17-s + 3.21·19-s − 1/5·25-s − 0.769·27-s + 0.696·33-s + 2.18·41-s + 0.304·43-s + 2/7·49-s − 0.560·51-s − 3.70·57-s + 2.60·59-s + 2.93·67-s − 3.27·73-s + 0.230·75-s + 5/9·81-s + 3.07·83-s + 1.69·89-s + 2.43·97-s − 0.603·99-s − 2.51·107-s − 3.57·113-s − 1.72·121-s − 2.52·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.924335194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.924335194\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72417736726020375123060781448, −7.10862606316740176491095490628, −6.86331177018920388447026866658, −6.32126457742274162742887328772, −5.63763740224640266713998216932, −5.57392449844244671802827644030, −5.24158903396702022407128081594, −4.81536414408362940263533105554, −4.13220745200772588524548445751, −3.73278236399044089261498793654, −3.15894889975583292757146607138, −2.63789286879136207172642804519, −1.94843284288093064344150345230, −0.906565211409185083613124704727, −0.829941819698449579623233485859,
0.829941819698449579623233485859, 0.906565211409185083613124704727, 1.94843284288093064344150345230, 2.63789286879136207172642804519, 3.15894889975583292757146607138, 3.73278236399044089261498793654, 4.13220745200772588524548445751, 4.81536414408362940263533105554, 5.24158903396702022407128081594, 5.57392449844244671802827644030, 5.63763740224640266713998216932, 6.32126457742274162742887328772, 6.86331177018920388447026866658, 7.10862606316740176491095490628, 7.72417736726020375123060781448
