| L(s) = 1 | − 5·2-s − 3-s + 12·4-s − 2·5-s + 5·6-s − 10·7-s − 20·8-s + 10·10-s − 12·12-s + 50·14-s + 2·15-s + 30·16-s − 10·17-s − 10·19-s − 24·20-s + 10·21-s − 16·23-s + 20·24-s + 5·25-s − 120·28-s + 10·29-s − 10·30-s − 45·32-s + 50·34-s + 20·35-s − 2·37-s + 50·38-s + ⋯ |
| L(s) = 1 | − 3.53·2-s − 0.577·3-s + 6·4-s − 0.894·5-s + 2.04·6-s − 3.77·7-s − 7.07·8-s + 3.16·10-s − 3.46·12-s + 13.3·14-s + 0.516·15-s + 15/2·16-s − 2.42·17-s − 2.29·19-s − 5.36·20-s + 2.18·21-s − 3.33·23-s + 4.08·24-s + 25-s − 22.6·28-s + 1.85·29-s − 1.82·30-s − 7.95·32-s + 8.57·34-s + 3.38·35-s − 0.328·37-s + 8.11·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 25 p T^{5} + 13 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 + 2 T - T^{2} - 12 T^{3} - 19 T^{4} - 12 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 10 T + 53 T^{2} + 200 T^{3} + 589 T^{4} + 200 p T^{5} + 53 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 10 T + 43 T^{2} + 200 T^{3} + 1029 T^{4} + 200 p T^{5} + 43 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 200 T^{3} + 1141 T^{4} + 200 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 200 p T^{5} + 31 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + 2 T - 33 T^{2} - 140 T^{3} + 941 T^{4} - 140 p T^{5} - 33 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 + 10 T + 19 T^{2} + 200 T^{3} + 2901 T^{4} + 200 p T^{5} + 19 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 240 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 420 p T^{5} - 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 20 T + 179 T^{2} + 1600 T^{3} + 15001 T^{4} + 1600 p T^{5} + 179 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 - 8 T - 7 T^{2} + 624 T^{3} - 4495 T^{4} + 624 p T^{5} - 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 20 T + 167 T^{2} + 1600 T^{3} + 17569 T^{4} + 1600 p T^{5} + 167 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 30 T + 461 T^{2} + 5400 T^{3} + 52861 T^{4} + 5400 p T^{5} + 461 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 20 T + 157 T^{2} - 1600 T^{3} + 19929 T^{4} - 1600 p T^{5} + 157 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 + 2 T - 93 T^{2} - 380 T^{3} + 8261 T^{4} - 380 p T^{5} - 93 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.911596981415651533460826170864, −8.619710990171056090780648214955, −8.488713369923183173420331367884, −8.316925767634156724823786351147, −7.919997173281999995308655437635, −7.76125293115082883441175143672, −7.36742244047383441975759703259, −7.08636131381566549613954010335, −6.65347699744298434320375230247, −6.60539953686446730429116473884, −6.53554551506523268766400769498, −6.37427797255033564209703097513, −5.95499319061068649806064755413, −5.76349084039087260304377611668, −5.63493262137862259144867806582, −4.57766661514690438307229868508, −4.34907605974048514928284053301, −4.30780903858763957364065878385, −4.06724442962360687649635614436, −3.18757331967181542271751575697, −3.16544726887110387131289330136, −2.91730361629439287662523984329, −2.57095219232293235532905053543, −1.74997079322915784401467708903, −1.59237093743335086277336395191, 0, 0, 0, 0,
1.59237093743335086277336395191, 1.74997079322915784401467708903, 2.57095219232293235532905053543, 2.91730361629439287662523984329, 3.16544726887110387131289330136, 3.18757331967181542271751575697, 4.06724442962360687649635614436, 4.30780903858763957364065878385, 4.34907605974048514928284053301, 4.57766661514690438307229868508, 5.63493262137862259144867806582, 5.76349084039087260304377611668, 5.95499319061068649806064755413, 6.37427797255033564209703097513, 6.53554551506523268766400769498, 6.60539953686446730429116473884, 6.65347699744298434320375230247, 7.08636131381566549613954010335, 7.36742244047383441975759703259, 7.76125293115082883441175143672, 7.919997173281999995308655437635, 8.316925767634156724823786351147, 8.488713369923183173420331367884, 8.619710990171056090780648214955, 8.911596981415651533460826170864
