| L(s) = 1 | − 8·13-s + 8·25-s − 8·37-s − 4·49-s + 56·73-s + 64·97-s − 80·109-s − 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 56·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 2.21·13-s + 8/5·25-s − 1.31·37-s − 4/7·49-s + 6.55·73-s + 6.49·97-s − 7.66·109-s − 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.002959012090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002959012090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 4 T^{2} + 51 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 28 T^{2} + 435 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 16 T^{2} + 210 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T^{2} - 477 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 76 T^{2} + 2499 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 64 T^{2} + 2274 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 82 T^{2} + 3171 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 20 T^{2} + 2379 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 116 T^{2} + 6762 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 32 T^{2} + 786 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 148 T^{2} + 11046 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 184 T^{2} + 14994 T^{4} + 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 244 T^{2} + 23754 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 148 T^{2} + 10995 T^{4} + 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 14 T + 168 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 292 T^{2} + 33690 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 280 T^{2} + 32946 T^{4} + 280 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 44 T^{2} + 14451 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.51538859148102533152092826613, −3.46178979947992732070894748754, −3.44578759374705839425872666617, −3.35471022711393060418866382213, −3.34262665675179605589494783285, −2.88611624971672746607677529264, −2.81956108348194474925158137926, −2.68614879571468400090336211969, −2.66754645395793662974942566070, −2.56246046782437254690314342004, −2.34553779936230555321595812090, −2.32318608644465386164523342410, −2.23523005326431426919173127247, −2.06159290881211731175440108645, −2.00542897389141079796858599105, −1.74205659174345718218193042066, −1.57611305206327523197396563173, −1.36616517361743664966463838635, −1.19357253812524673071411224814, −1.14805654285178746984617934702, −1.08871502657599810322326160921, −0.72082339777811511695306963458, −0.48258512232289641034345248407, −0.39587385869476712972494502592, −0.00507171130444320884595804447,
0.00507171130444320884595804447, 0.39587385869476712972494502592, 0.48258512232289641034345248407, 0.72082339777811511695306963458, 1.08871502657599810322326160921, 1.14805654285178746984617934702, 1.19357253812524673071411224814, 1.36616517361743664966463838635, 1.57611305206327523197396563173, 1.74205659174345718218193042066, 2.00542897389141079796858599105, 2.06159290881211731175440108645, 2.23523005326431426919173127247, 2.32318608644465386164523342410, 2.34553779936230555321595812090, 2.56246046782437254690314342004, 2.66754645395793662974942566070, 2.68614879571468400090336211969, 2.81956108348194474925158137926, 2.88611624971672746607677529264, 3.34262665675179605589494783285, 3.35471022711393060418866382213, 3.44578759374705839425872666617, 3.46178979947992732070894748754, 3.51538859148102533152092826613
Plot not available for L-functions of degree greater than 10.
