| L(s) = 1 | − 3·4-s − 6·7-s − 6·11-s + 6·16-s − 3·25-s + 18·28-s − 2·37-s + 12·41-s + 18·44-s + 24·47-s + 5·49-s − 6·53-s − 10·64-s + 20·71-s − 26·73-s + 36·77-s − 14·83-s + 9·100-s − 8·101-s − 50·107-s − 36·112-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 6·148-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 2.26·7-s − 1.80·11-s + 3/2·16-s − 3/5·25-s + 3.40·28-s − 0.328·37-s + 1.87·41-s + 2.71·44-s + 3.50·47-s + 5/7·49-s − 0.824·53-s − 5/4·64-s + 2.37·71-s − 3.04·73-s + 4.10·77-s − 1.53·83-s + 9/10·100-s − 0.796·101-s − 4.83·107-s − 3.40·112-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.493·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5034741223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5034741223\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{3} \) |
| 37 | \( 1 + 2 T + 31 T^{2} + 116 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 7 | \( ( 1 + 3 T + 11 T^{2} + 26 T^{3} + 11 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 + 3 T + 23 T^{2} + 58 T^{3} + 23 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 25 T^{2} + 119 T^{4} + 1026 T^{6} + 119 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 19 T^{2} + 771 T^{4} + 8674 T^{6} + 771 p^{2} T^{8} + 19 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 61 T^{2} + 1727 T^{4} - 35238 T^{6} + 1727 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 25 T^{2} + 419 T^{4} - 774 T^{6} + 419 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( 1 - 66 T^{2} + 2111 T^{4} - 63356 T^{6} + 2111 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 43 | \( 1 - 94 T^{2} + 6055 T^{4} - 331108 T^{6} + 6055 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 12 T + 137 T^{2} - 952 T^{3} + 137 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 + 3 T + 149 T^{2} + 310 T^{3} + 149 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 170 T^{2} + 15207 T^{4} - 990668 T^{6} + 15207 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{3} \) |
| 67 | \( ( 1 - 7 T^{2} - 256 T^{3} - 7 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 10 T + 21 T^{2} + 468 T^{3} + 21 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 13 T + 191 T^{2} + 1446 T^{3} + 191 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 - 226 T^{2} + 28063 T^{4} - 2568124 T^{6} + 28063 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 7 T + 181 T^{2} + 810 T^{3} + 181 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 225 T^{2} + 27615 T^{4} - 2438334 T^{6} + 27615 p^{2} T^{8} - 225 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 430 T^{2} + 84863 T^{4} - 10184484 T^{6} + 84863 p^{2} T^{8} - 430 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.27466768848111523994091600264, −4.21594424297062005361230934068, −4.15653786900666020914678084547, −4.11722507872913479241489191305, −3.85537860899494952358593414633, −3.84191628996425138598798106390, −3.79564800919907901034906476791, −3.32726519410233554423272152097, −3.17170236955181322779549761860, −3.11481221499821211685596745131, −2.85834565546965752520392073896, −2.79068267126582215474939434480, −2.77953127010587209090532147899, −2.63307335500337638059311192294, −2.57015074008354329430528773611, −2.07927365534876293991610865485, −1.98851913494677918516264586813, −1.77605053565689213495043046846, −1.57784748886960220896653656684, −1.23588283155390227869120259849, −1.20115089309241207821207885068, −0.76510851797553392525648481286, −0.54357798527110780444767136925, −0.35618708706655334385026727382, −0.16903558315963790716294049669,
0.16903558315963790716294049669, 0.35618708706655334385026727382, 0.54357798527110780444767136925, 0.76510851797553392525648481286, 1.20115089309241207821207885068, 1.23588283155390227869120259849, 1.57784748886960220896653656684, 1.77605053565689213495043046846, 1.98851913494677918516264586813, 2.07927365534876293991610865485, 2.57015074008354329430528773611, 2.63307335500337638059311192294, 2.77953127010587209090532147899, 2.79068267126582215474939434480, 2.85834565546965752520392073896, 3.11481221499821211685596745131, 3.17170236955181322779549761860, 3.32726519410233554423272152097, 3.79564800919907901034906476791, 3.84191628996425138598798106390, 3.85537860899494952358593414633, 4.11722507872913479241489191305, 4.15653786900666020914678084547, 4.21594424297062005361230934068, 4.27466768848111523994091600264
Plot not available for L-functions of degree greater than 10.
