| L(s) = 1 | + 4·2-s + 4·3-s + 8·4-s − 4·5-s + 16·6-s + 8·8-s + 11·9-s − 16·10-s + 6·11-s + 32·12-s − 12·13-s − 16·15-s − 4·16-s + 6·17-s + 44·18-s + 12·19-s − 32·20-s + 24·22-s + 12·23-s + 32·24-s + 5·25-s − 48·26-s + 20·27-s − 4·29-s − 64·30-s − 18·31-s − 32·32-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 4·4-s − 1.78·5-s + 6.53·6-s + 2.82·8-s + 11/3·9-s − 5.05·10-s + 1.80·11-s + 9.23·12-s − 3.32·13-s − 4.13·15-s − 16-s + 1.45·17-s + 10.3·18-s + 2.75·19-s − 7.15·20-s + 5.11·22-s + 2.50·23-s + 6.53·24-s + 25-s − 9.41·26-s + 3.84·27-s − 0.742·29-s − 11.6·30-s − 3.23·31-s − 5.65·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.34834608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.34834608\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 144 T^{3} + 587 T^{4} - 144 p T^{5} + 36 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 45 T^{2} + 84 T^{3} - 1276 T^{4} + 84 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 47 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 18 T + 172 T^{2} + 1152 T^{3} + 6483 T^{4} + 1152 p T^{5} + 172 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 300 T^{3} + 551 T^{4} + 300 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 276 T^{3} + 4223 T^{4} - 276 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 90 T^{2} + 528 T^{3} - 8377 T^{4} + 528 p T^{5} + 90 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 24 T + 180 T^{2} - 204 T^{3} - 9145 T^{4} - 204 p T^{5} + 180 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 6 p T^{2} - 3888 T^{3} + 34091 T^{4} - 3888 p T^{5} + 6 p^{3} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 47 T^{2} + 234 T^{3} + 972 T^{4} + 234 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 117 T^{2} + 996 T^{3} + 7052 T^{4} + 996 p T^{5} + 117 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 24 T + 274 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 26 T + 290 T^{2} + 1776 T^{3} + 10223 T^{4} + 1776 p T^{5} + 290 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 24 T + 362 T^{2} - 4080 T^{3} + 37827 T^{4} - 4080 p T^{5} + 362 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 84 T^{3} - 4369 T^{4} - 84 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 281 T^{2} - 2808 T^{3} + 27840 T^{4} - 2808 p T^{5} + 281 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 7008 T^{3} + 81038 T^{4} + 7008 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.856150741884632480976051535263, −8.013393780779249849632467863231, −7.985806684140267551315960356137, −7.74755342338894154378820966290, −7.41270832456188885269580035708, −7.34015116366196970148201708593, −7.18891416598227618562113383834, −6.91478729484676280178054894713, −6.74540192067330739695182476977, −6.29252902301538921845250620458, −5.63524496723586120461980288714, −5.44865415767917462709124295649, −5.10573423713970040928633239350, −5.06651400597055332186517501619, −4.78478021936815025531800260754, −4.35721761208681542601523186705, −3.90394222983684546146223926867, −3.82254502998517373614979056898, −3.51373938271878839443148989183, −3.38019176946268871953115662208, −3.21494422213599849990499540359, −2.68270785133712729551857699180, −2.30885854148395111787779706013, −1.98976639421233728469804954924, −1.07820900346388312666851896048,
1.07820900346388312666851896048, 1.98976639421233728469804954924, 2.30885854148395111787779706013, 2.68270785133712729551857699180, 3.21494422213599849990499540359, 3.38019176946268871953115662208, 3.51373938271878839443148989183, 3.82254502998517373614979056898, 3.90394222983684546146223926867, 4.35721761208681542601523186705, 4.78478021936815025531800260754, 5.06651400597055332186517501619, 5.10573423713970040928633239350, 5.44865415767917462709124295649, 5.63524496723586120461980288714, 6.29252902301538921845250620458, 6.74540192067330739695182476977, 6.91478729484676280178054894713, 7.18891416598227618562113383834, 7.34015116366196970148201708593, 7.41270832456188885269580035708, 7.74755342338894154378820966290, 7.985806684140267551315960356137, 8.013393780779249849632467863231, 8.856150741884632480976051535263
