| L(s) = 1 | + 3-s + 4-s + 9-s + 12-s − 4·13-s + 16-s + 8·19-s − 6·25-s + 27-s + 16·31-s + 36-s − 4·37-s − 4·39-s + 24·43-s + 48-s − 14·49-s − 4·52-s + 8·57-s − 20·61-s + 64-s − 24·67-s + 20·73-s − 6·75-s + 8·76-s − 16·79-s + 81-s + 16·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1/3·9-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 1.83·19-s − 6/5·25-s + 0.192·27-s + 2.87·31-s + 1/6·36-s − 0.657·37-s − 0.640·39-s + 3.65·43-s + 0.144·48-s − 2·49-s − 0.554·52-s + 1.05·57-s − 2.56·61-s + 1/8·64-s − 2.93·67-s + 2.34·73-s − 0.692·75-s + 0.917·76-s − 1.80·79-s + 1/9·81-s + 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.697448101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697448101\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−10.52566566553203306448675052964, −9.723242141956922614700830245165, −9.570394810938145814587700544156, −9.104444876573252449353403617805, −8.000937863938982294341146593643, −7.958882066985845549410362974675, −7.36440720073199929079936076774, −6.79533660929922109224513153399, −6.05886413503827459123865903257, −5.51457882926074302977186886250, −4.66119827871442481374467622682, −4.13875090005906653574798019221, −2.92323665285260202872977027719, −2.76309813409348720224135008444, −1.43438626728056627432143840048,
1.43438626728056627432143840048, 2.76309813409348720224135008444, 2.92323665285260202872977027719, 4.13875090005906653574798019221, 4.66119827871442481374467622682, 5.51457882926074302977186886250, 6.05886413503827459123865903257, 6.79533660929922109224513153399, 7.36440720073199929079936076774, 7.958882066985845549410362974675, 8.000937863938982294341146593643, 9.104444876573252449353403617805, 9.570394810938145814587700544156, 9.723242141956922614700830245165, 10.52566566553203306448675052964
