| L(s) = 1 | + 2·5-s − 8·7-s + 12·11-s − 2·13-s + 4·17-s + 6·23-s + 4·25-s + 4·29-s − 8·31-s − 16·35-s − 20·37-s + 8·41-s + 16·43-s + 26·49-s − 2·53-s + 24·55-s − 10·59-s − 10·61-s − 4·65-s + 12·67-s + 20·71-s + 2·73-s − 96·77-s + 12·79-s + 40·83-s + 8·85-s + 22·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 3.02·7-s + 3.61·11-s − 0.554·13-s + 0.970·17-s + 1.25·23-s + 4/5·25-s + 0.742·29-s − 1.43·31-s − 2.70·35-s − 3.28·37-s + 1.24·41-s + 2.43·43-s + 26/7·49-s − 0.274·53-s + 3.23·55-s − 1.30·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s + 2.37·71-s + 0.234·73-s − 10.9·77-s + 1.35·79-s + 4.39·83-s + 0.867·85-s + 2.33·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.576231493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.576231493\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T + 12 T^{3} - 29 T^{4} + 12 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T + 5 T^{2} - 54 T^{3} - 220 T^{4} - 54 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} + 96 T^{3} - 461 T^{4} + 96 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 12 T^{2} - 12 T^{3} + 947 T^{4} - 12 p T^{5} - 12 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 18 T^{2} + 96 T^{3} - 149 T^{4} + 96 p T^{5} - 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 16 T + 113 T^{2} - 912 T^{3} + 7592 T^{4} - 912 p T^{5} + 113 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 66 T^{2} + 2147 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 2 T - 40 T^{2} - 124 T^{3} - 1085 T^{4} - 124 p T^{5} - 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 10 T - 36 T^{2} + 180 T^{3} + 9115 T^{4} + 180 p T^{5} - 36 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 37 T^{2} + 324 T^{3} - 2688 T^{4} + 324 p T^{5} + 37 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 10 T + 29 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 2 T - 115 T^{2} + 54 T^{3} + 8540 T^{4} + 54 p T^{5} - 115 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12 T - 43 T^{2} - 348 T^{3} + 17352 T^{4} - 348 p T^{5} - 43 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 22 T + 192 T^{2} - 2508 T^{3} + 34267 T^{4} - 2508 p T^{5} + 192 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 - 82 T^{2} - 2685 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.46084989563486434030339094334, −6.05595048424363759793799952646, −6.00866136520050950274006337461, −5.81179048132784882460581486983, −5.60678657260222155454031640246, −5.11771102729348820107478581295, −5.01882235295276316494028019605, −4.84586669898572655767927281304, −4.76174945468381722555895304265, −4.29909661260733702719300595313, −4.01227249251537783899558765503, −3.68452226875966656797923556494, −3.64647028854137521624095169115, −3.51818685329322313802497192121, −3.33894837675064523450686012690, −3.19064090053637066300459285198, −2.90388740464471365206284279934, −2.53393378047613483728375356270, −2.01504775934323327894855548219, −1.96701204361077581878483796526, −1.95776081374080646288347216705, −1.20886435402743005694608235350, −1.02889705459625545882793691640, −0.60408423926781223813857666800, −0.57867814657421694046019101469,
0.57867814657421694046019101469, 0.60408423926781223813857666800, 1.02889705459625545882793691640, 1.20886435402743005694608235350, 1.95776081374080646288347216705, 1.96701204361077581878483796526, 2.01504775934323327894855548219, 2.53393378047613483728375356270, 2.90388740464471365206284279934, 3.19064090053637066300459285198, 3.33894837675064523450686012690, 3.51818685329322313802497192121, 3.64647028854137521624095169115, 3.68452226875966656797923556494, 4.01227249251537783899558765503, 4.29909661260733702719300595313, 4.76174945468381722555895304265, 4.84586669898572655767927281304, 5.01882235295276316494028019605, 5.11771102729348820107478581295, 5.60678657260222155454031640246, 5.81179048132784882460581486983, 6.00866136520050950274006337461, 6.05595048424363759793799952646, 6.46084989563486434030339094334
