| L(s) = 1 | − 2·3-s + 4·7-s + 9-s + 4·13-s − 8·19-s − 8·21-s + 25-s + 4·27-s − 8·31-s + 4·37-s − 8·39-s − 20·43-s − 2·49-s + 16·57-s + 4·61-s + 4·63-s + 4·67-s + 4·73-s − 2·75-s + 16·79-s − 11·81-s + 16·91-s + 16·93-s + 4·97-s + 28·103-s + 4·109-s − 8·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.10·13-s − 1.83·19-s − 1.74·21-s + 1/5·25-s + 0.769·27-s − 1.43·31-s + 0.657·37-s − 1.28·39-s − 3.04·43-s − 2/7·49-s + 2.11·57-s + 0.512·61-s + 0.503·63-s + 0.488·67-s + 0.468·73-s − 0.230·75-s + 1.80·79-s − 1.22·81-s + 1.67·91-s + 1.65·93-s + 0.406·97-s + 2.75·103-s + 0.383·109-s − 0.759·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6180626677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6180626677\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−12.64919926214782333798361971460, −11.66835822739294093715813029994, −11.50854453196989171707939391081, −10.83473950301065015207539180976, −10.66869217043212474435745329357, −9.749524176059453166876684221310, −8.552217550781204179646223792184, −8.536040373596851277524077678909, −7.64589778281564359800589548721, −6.57891116465648258947670054106, −6.21444115782927123192608967686, −5.19647848534431193907436500640, −4.78130792717525308450176413839, −3.71119793411816721603957340078, −1.81793015252092636076156145980,
1.81793015252092636076156145980, 3.71119793411816721603957340078, 4.78130792717525308450176413839, 5.19647848534431193907436500640, 6.21444115782927123192608967686, 6.57891116465648258947670054106, 7.64589778281564359800589548721, 8.536040373596851277524077678909, 8.552217550781204179646223792184, 9.749524176059453166876684221310, 10.66869217043212474435745329357, 10.83473950301065015207539180976, 11.50854453196989171707939391081, 11.66835822739294093715813029994, 12.64919926214782333798361971460
