| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s + 2·23-s − 4·25-s + 36-s + 49-s − 2·53-s − 4·69-s + 8·75-s + 2·92-s − 4·100-s + 2·103-s − 2·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 2·147-s + 149-s + 151-s + 157-s + 4·159-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s + 2·23-s − 4·25-s + 36-s + 49-s − 2·53-s − 4·69-s + 8·75-s + 2·92-s − 4·100-s + 2·103-s − 2·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 2·147-s + 149-s + 151-s + 157-s + 4·159-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2495209845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2495209845\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 3 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.60484534804742234873286411285, −6.29692062094946694710022732024, −6.27297140662719251114481611182, −6.25646414203093292681187990034, −6.02684996587430093022854228675, −5.74092703204870365036091661391, −5.53395571496240052042756517294, −5.39414066333506429847196884150, −5.19160015289070285605485097175, −4.90487023365986213254722904610, −4.79186006953995379447151004771, −4.55757157882547882637906045316, −3.99550743835623145647912398860, −3.97162095399606170653136423171, −3.84202967344876493561060808719, −3.48053015922686279154758778667, −3.21031578507652648803809475803, −2.75936600320791000665323816245, −2.71681443846355516991990598829, −2.32744956284831945345976316818, −1.97554491217688452470336438071, −1.86848457489357647895693523620, −1.32682616483191310955730332663, −1.14676397370918974089592620189, −0.31049275573800656140590215459,
0.31049275573800656140590215459, 1.14676397370918974089592620189, 1.32682616483191310955730332663, 1.86848457489357647895693523620, 1.97554491217688452470336438071, 2.32744956284831945345976316818, 2.71681443846355516991990598829, 2.75936600320791000665323816245, 3.21031578507652648803809475803, 3.48053015922686279154758778667, 3.84202967344876493561060808719, 3.97162095399606170653136423171, 3.99550743835623145647912398860, 4.55757157882547882637906045316, 4.79186006953995379447151004771, 4.90487023365986213254722904610, 5.19160015289070285605485097175, 5.39414066333506429847196884150, 5.53395571496240052042756517294, 5.74092703204870365036091661391, 6.02684996587430093022854228675, 6.25646414203093292681187990034, 6.27297140662719251114481611182, 6.29692062094946694710022732024, 6.60484534804742234873286411285
