| L(s) = 1 | + 3-s + 9-s + 4·11-s + 2·17-s + 8·19-s + 6·25-s + 27-s + 4·33-s − 6·41-s + 12·43-s + 2·49-s + 2·51-s + 8·57-s − 4·59-s − 8·67-s + 4·73-s + 6·75-s + 81-s − 12·83-s + 2·89-s − 16·97-s + 4·99-s − 20·107-s + 10·113-s − 10·121-s − 6·123-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.485·17-s + 1.83·19-s + 6/5·25-s + 0.192·27-s + 0.696·33-s − 0.937·41-s + 1.82·43-s + 2/7·49-s + 0.280·51-s + 1.05·57-s − 0.520·59-s − 0.977·67-s + 0.468·73-s + 0.692·75-s + 1/9·81-s − 1.31·83-s + 0.211·89-s − 1.62·97-s + 0.402·99-s − 1.93·107-s + 0.940·113-s − 0.909·121-s − 0.541·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.937281826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.937281826\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.662472690379954740076955391737, −8.136815330843858226575976911329, −7.67658937210905477332269099463, −7.18953213215964577524164161482, −6.91790924585150685734277023890, −6.34751711026311015645731535279, −5.73209898252443192890277929174, −5.33664331370749068673432018381, −4.70527905596991667445807995872, −4.15559252554397655131772081855, −3.62054906284588403475623971058, −3.07032264912637567606068890793, −2.60552801541723321287618387213, −1.52543873211525956626386179618, −1.03680801872388207048737512720,
1.03680801872388207048737512720, 1.52543873211525956626386179618, 2.60552801541723321287618387213, 3.07032264912637567606068890793, 3.62054906284588403475623971058, 4.15559252554397655131772081855, 4.70527905596991667445807995872, 5.33664331370749068673432018381, 5.73209898252443192890277929174, 6.34751711026311015645731535279, 6.91790924585150685734277023890, 7.18953213215964577524164161482, 7.67658937210905477332269099463, 8.136815330843858226575976911329, 8.662472690379954740076955391737
