| L(s) = 1 | − 8·2-s − 18·3-s + 16·4-s + 108·5-s + 144·6-s + 128·8-s + 81·9-s − 864·10-s + 124·11-s − 288·12-s − 1.44e3·13-s − 1.94e3·15-s − 1.02e3·16-s + 612·17-s − 648·18-s + 2.08e3·19-s + 1.72e3·20-s − 992·22-s − 772·23-s − 2.30e3·24-s + 9.11e3·25-s + 1.15e4·26-s + 1.45e3·27-s − 9.18e3·29-s + 1.55e4·30-s + 9.79e3·31-s + 2.04e3·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.93·5-s + 1.63·6-s + 0.707·8-s + 1/3·9-s − 2.73·10-s + 0.308·11-s − 0.577·12-s − 2.36·13-s − 2.23·15-s − 16-s + 0.513·17-s − 0.471·18-s + 1.32·19-s + 0.965·20-s − 0.436·22-s − 0.304·23-s − 0.816·24-s + 2.91·25-s + 3.34·26-s + 0.384·27-s − 2.02·29-s + 3.15·30-s + 1.83·31-s + 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.02245776062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02245776062\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 108 T + 2548 T^{2} - 309528 T^{3} + 36885831 T^{4} - 309528 p^{5} T^{5} + 2548 p^{10} T^{6} - 108 p^{15} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 124 T - 309922 T^{2} - 396304 T^{3} + 77405044027 T^{4} - 396304 p^{5} T^{5} - 309922 p^{10} T^{6} - 124 p^{15} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 720 T + 716504 T^{2} + 720 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 36 p T + 1804 p^{2} T^{2} + 372024 p^{3} T^{3} - 30246753 p^{4} T^{4} + 372024 p^{8} T^{5} + 1804 p^{12} T^{6} - 36 p^{16} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2088 T + 556338 T^{2} + 2398677696 T^{3} - 2460326815621 T^{4} + 2398677696 p^{5} T^{5} + 556338 p^{10} T^{6} - 2088 p^{15} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 772 T - 10740250 T^{2} - 1186140944 T^{3} + 83247732642595 T^{4} - 1186140944 p^{5} T^{5} - 10740250 p^{10} T^{6} + 772 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4592 T + 40196882 T^{2} + 4592 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 9792 T + 14787274 T^{2} - 233418640896 T^{3} + 3011874863670435 T^{4} - 233418640896 p^{5} T^{5} + 14787274 p^{10} T^{6} - 9792 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 5992 T - 15210458 T^{2} + 524739764864 T^{3} - 4685619883270613 T^{4} + 524739764864 p^{5} T^{5} - 15210458 p^{10} T^{6} - 5992 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 20196 T + 324853604 T^{2} + 20196 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 1136 T + 157222710 T^{2} + 1136 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 36936 T + 589263706 T^{2} - 11683387791936 T^{3} + 235144596256445379 T^{4} - 11683387791936 p^{5} T^{5} + 589263706 p^{10} T^{6} - 36936 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16708 T - 625270846 T^{2} - 1136764267792 T^{3} + 526360426149647899 T^{4} - 1136764267792 p^{5} T^{5} - 625270846 p^{10} T^{6} - 16708 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 74592 T + 2791107058 T^{2} - 100177862190336 T^{3} + 3199817708207322699 T^{4} - 100177862190336 p^{5} T^{5} + 2791107058 p^{10} T^{6} - 74592 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18648 T - 1327249032 T^{2} - 264720779568 T^{3} + 1798565230547602391 T^{4} - 264720779568 p^{5} T^{5} - 1327249032 p^{10} T^{6} + 18648 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 67344 T + 863160538 T^{2} + 65445140560896 T^{3} + 5538034550194529403 T^{4} + 65445140560896 p^{5} T^{5} + 863160538 p^{10} T^{6} + 67344 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 76548 T + 3345343306 T^{2} - 76548 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 648 p T - 2287961424 T^{2} - 245907351792 p T^{3} + 11544852791231961263 T^{4} - 245907351792 p^{6} T^{5} - 2287961424 p^{10} T^{6} - 648 p^{16} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 140656 T + 9482370754 T^{2} + 583388592930304 T^{3} + 35780739024807847459 T^{4} + 583388592930304 p^{5} T^{5} + 9482370754 p^{10} T^{6} + 140656 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 94104 T + 8432170262 T^{2} + 94104 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 17604 T - 7299865748 T^{2} - 62641234487736 T^{3} + 24996322648768572111 T^{4} - 62641234487736 p^{5} T^{5} - 7299865748 p^{10} T^{6} + 17604 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 85176 T + 11802571296 T^{2} + 85176 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−7.70727137067394487127927241462, −7.23359244430428747218196624152, −7.16303003550840066179163364602, −6.95685814746379592541523498874, −6.77333228664798831982057537074, −6.54066614978360558478846212700, −5.89305400142353704721752748212, −5.88214388309627707603379116910, −5.42929931438859363164514173476, −5.42042055800965840966283142916, −5.21880537331044452674625487182, −4.85252359248570732967000159265, −4.63544094189059521318662516656, −4.10646509164222393857938209105, −3.97246458168992858530276665448, −3.28212807909080380237813309844, −2.98171201421556822787595819215, −2.62665226951817819904782040170, −2.36027543222915141452321961214, −1.87309944060745736984388791615, −1.74815761770332222277528006705, −1.18334164418394292871357046483, −0.920144686825478246991350279564, −0.64728673517564259845963850735, −0.03226152419804914656898436522,
0.03226152419804914656898436522, 0.64728673517564259845963850735, 0.920144686825478246991350279564, 1.18334164418394292871357046483, 1.74815761770332222277528006705, 1.87309944060745736984388791615, 2.36027543222915141452321961214, 2.62665226951817819904782040170, 2.98171201421556822787595819215, 3.28212807909080380237813309844, 3.97246458168992858530276665448, 4.10646509164222393857938209105, 4.63544094189059521318662516656, 4.85252359248570732967000159265, 5.21880537331044452674625487182, 5.42042055800965840966283142916, 5.42929931438859363164514173476, 5.88214388309627707603379116910, 5.89305400142353704721752748212, 6.54066614978360558478846212700, 6.77333228664798831982057537074, 6.95685814746379592541523498874, 7.16303003550840066179163364602, 7.23359244430428747218196624152, 7.70727137067394487127927241462
