| L(s) = 1 | + 2-s + 5-s + 2·7-s − 8-s − 9-s + 10-s − 2·11-s + 2·13-s + 2·14-s − 16-s − 18-s − 19-s − 2·22-s − 23-s + 2·26-s + 2·35-s + 2·37-s − 38-s − 40-s + 41-s − 45-s − 46-s + 2·47-s + 49-s − 53-s − 2·55-s − 2·56-s + ⋯ |
| L(s) = 1 | + 2-s + 5-s + 2·7-s − 8-s − 9-s + 10-s − 2·11-s + 2·13-s + 2·14-s − 16-s − 18-s − 19-s − 2·22-s − 23-s + 2·26-s + 2·35-s + 2·37-s − 38-s − 40-s + 41-s − 45-s − 46-s + 2·47-s + 49-s − 53-s − 2·55-s − 2·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.607498613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.607498613\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.93485441171755481527707135520, −10.48250121262372326865424083650, −10.06825674519893421042825104863, −9.394745800195968114513699742530, −8.830312691135980401438075322715, −8.727327446560190403340937227162, −8.058704218817334551105236586684, −7.88324086271870167173093584438, −7.59851929498179832566232844072, −6.31742644072130962741723524899, −6.18683273596882901435283575561, −5.74952792046197916440694269206, −5.53957220539833212944099355730, −4.82679929257341686661846767837, −4.60651403833724601930028635338, −4.01744006617182300204703109441, −3.33634997677515689541557517985, −2.40768412127471461183800674231, −2.40058467866563171112762691553, −1.37604636577870496423637973947,
1.37604636577870496423637973947, 2.40058467866563171112762691553, 2.40768412127471461183800674231, 3.33634997677515689541557517985, 4.01744006617182300204703109441, 4.60651403833724601930028635338, 4.82679929257341686661846767837, 5.53957220539833212944099355730, 5.74952792046197916440694269206, 6.18683273596882901435283575561, 6.31742644072130962741723524899, 7.59851929498179832566232844072, 7.88324086271870167173093584438, 8.058704218817334551105236586684, 8.727327446560190403340937227162, 8.830312691135980401438075322715, 9.394745800195968114513699742530, 10.06825674519893421042825104863, 10.48250121262372326865424083650, 10.93485441171755481527707135520
