| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s + 4·8-s − 4·10-s − 2·11-s + 16·13-s − 12·16-s + 10·17-s − 2·19-s + 4·20-s + 4·22-s − 6·23-s + 25-s − 32·26-s − 4·29-s − 6·31-s + 16·32-s − 20·34-s − 4·37-s + 4·38-s + 8·40-s − 4·41-s − 8·43-s − 4·44-s + 12·46-s + 4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s + 1.41·8-s − 1.26·10-s − 0.603·11-s + 4.43·13-s − 3·16-s + 2.42·17-s − 0.458·19-s + 0.894·20-s + 0.852·22-s − 1.25·23-s + 1/5·25-s − 6.27·26-s − 0.742·29-s − 1.07·31-s + 2.82·32-s − 3.42·34-s − 0.657·37-s + 0.648·38-s + 1.26·40-s − 0.624·41-s − 1.21·43-s − 0.603·44-s + 1.76·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.775617026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.775617026\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T + p T^{2} )^{2}( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 4 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 8 T + 3 p T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 10 T + 44 T^{2} - 220 T^{3} + 1147 T^{4} - 220 p T^{5} + 44 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 23 T^{2} - 22 T^{3} + 292 T^{4} - 22 p T^{5} - 23 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 18 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 35 T^{2} - 92 T^{3} + 640 T^{4} - 92 p T^{5} - 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 80 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 14 T^{2} + 416 T^{3} - 3653 T^{4} + 416 p T^{5} + 14 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 20 p T^{5} - 16 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 49 T^{2} + 468 T^{3} - 5112 T^{4} + 468 p T^{5} + 49 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T - 23 T^{2} - 472 T^{3} - 2432 T^{4} - 472 p T^{5} - 23 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 594 T^{3} - 5604 T^{4} - 594 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 4 T^{2} + 828 T^{3} - 9525 T^{4} + 828 p T^{5} - 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.543032861985796281315126504284, −8.329964424442525961963791499918, −7.889191619018634660923270362022, −7.88158657293636644475976775724, −7.70706358175997892451873797249, −7.39519618523145091264564336695, −6.76110145095374915212952486085, −6.75437665933937998662421897053, −6.68581478628377410754662477614, −6.01347124749460117831238023397, −5.80560887138700782961388707185, −5.72282529615036722653565143202, −5.59934437717660654300007051289, −5.12005318141173849454984882081, −4.66133608573315113832298876367, −4.37643004357466078731241563937, −3.96687288362390057868290466107, −3.61171018310801935757337566004, −3.36641935276834435815412929471, −3.32749106160138390791360595062, −2.41746114642680862324084948007, −1.83928850010978849005790941824, −1.56172082471982527027468279590, −1.48615949979414703311502517711, −0.871949197295696055362585111969,
0.871949197295696055362585111969, 1.48615949979414703311502517711, 1.56172082471982527027468279590, 1.83928850010978849005790941824, 2.41746114642680862324084948007, 3.32749106160138390791360595062, 3.36641935276834435815412929471, 3.61171018310801935757337566004, 3.96687288362390057868290466107, 4.37643004357466078731241563937, 4.66133608573315113832298876367, 5.12005318141173849454984882081, 5.59934437717660654300007051289, 5.72282529615036722653565143202, 5.80560887138700782961388707185, 6.01347124749460117831238023397, 6.68581478628377410754662477614, 6.75437665933937998662421897053, 6.76110145095374915212952486085, 7.39519618523145091264564336695, 7.70706358175997892451873797249, 7.88158657293636644475976775724, 7.889191619018634660923270362022, 8.329964424442525961963791499918, 8.543032861985796281315126504284
