| L(s) = 1 | + 6·2-s + 21·4-s + 56·8-s − 3·9-s − 2·11-s + 126·16-s − 18·18-s − 12·22-s + 20·23-s − 11·25-s + 6·29-s + 252·32-s − 63·36-s + 28·37-s − 6·43-s − 42·44-s + 120·46-s − 66·50-s + 6·53-s + 36·58-s + 462·64-s + 12·67-s + 8·71-s − 168·72-s + 168·74-s + 18·79-s − 2·81-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 21/2·4-s + 19.7·8-s − 9-s − 0.603·11-s + 63/2·16-s − 4.24·18-s − 2.55·22-s + 4.17·23-s − 2.19·25-s + 1.11·29-s + 44.5·32-s − 10.5·36-s + 4.60·37-s − 0.914·43-s − 6.33·44-s + 17.6·46-s − 9.33·50-s + 0.824·53-s + 4.72·58-s + 57.7·64-s + 1.46·67-s + 0.949·71-s − 19.7·72-s + 19.5·74-s + 2.02·79-s − 2/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(278.3044441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(278.3044441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{6} \) |
| 7 | \( 1 \) |
| 29 | \( ( 1 - T )^{6} \) |
| good | 3 | \( 1 + p T^{2} + 11 T^{4} + 34 T^{6} + 11 p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 + 11 T^{2} + 47 T^{4} + 162 T^{6} + 47 p^{2} T^{8} + 11 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + T - 3 T^{2} - 18 T^{3} - 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 59 T^{2} + 123 p T^{4} + 26018 T^{6} + 123 p^{3} T^{8} + 59 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 74 T^{2} + 2551 T^{4} + 53660 T^{6} + 2551 p^{2} T^{8} + 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 62 T^{2} + 2231 T^{4} + 50532 T^{6} + 2231 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 10 T + 3 p T^{2} - 348 T^{3} + 3 p^{2} T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 87 T^{2} + 2951 T^{4} + 75130 T^{6} + 2951 p^{2} T^{8} + 87 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 14 T + 99 T^{2} - 548 T^{3} + 99 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 22 T^{2} + 5063 T^{4} - 73956 T^{6} + 5063 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 3 T + 113 T^{2} + 242 T^{3} + 113 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 135 T^{2} + 10247 T^{4} + 531802 T^{6} + 10247 p^{2} T^{8} + 135 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 3 T + 143 T^{2} - 298 T^{3} + 143 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 142 T^{2} + 12991 T^{4} + 948820 T^{6} + 12991 p^{2} T^{8} + 142 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 + 234 T^{2} + 27767 T^{4} + 2093836 T^{6} + 27767 p^{2} T^{8} + 234 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 6 T + 137 T^{2} - 676 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 4 T + 77 T^{2} - 1016 T^{3} + 77 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 410 T^{2} + 71879 T^{4} + 6902460 T^{6} + 71879 p^{2} T^{8} + 410 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 9 T + 245 T^{2} - 1390 T^{3} + 245 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 14 T^{2} + 4975 T^{4} + 547220 T^{6} + 4975 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 238 T^{2} + 13831 T^{4} + 101428 T^{6} + 13831 p^{2} T^{8} + 238 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 118 T^{2} + 13911 T^{4} - 566884 T^{6} + 13911 p^{2} T^{8} - 118 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.55955623812166782964169344904, −4.40466175570534429104058319366, −4.22537836108452746765275349877, −4.11042014415655920228918629503, −4.10356310620724732544164225161, −3.91028995471639259639817350450, −3.69888710271400459259008466098, −3.48573324234880655266363703540, −3.31168337844983040532412031550, −3.15610927704290823617717056798, −3.14788977645547624961585925402, −3.07359764668139625014702387533, −2.88508412640969975489898991249, −2.66074575764918798416278417555, −2.40729478207813297329082108844, −2.31782102706133807929910935896, −2.31466718753852260130419170255, −2.11009001612084157111746428858, −1.99446890846902144301053859310, −1.53116527085277678459756154439, −1.37409508133686273196265761907, −1.10987035545176256347733063743, −0.834557481861132548908837581887, −0.77282445146525268129172903861, −0.46792106080663830312129838989,
0.46792106080663830312129838989, 0.77282445146525268129172903861, 0.834557481861132548908837581887, 1.10987035545176256347733063743, 1.37409508133686273196265761907, 1.53116527085277678459756154439, 1.99446890846902144301053859310, 2.11009001612084157111746428858, 2.31466718753852260130419170255, 2.31782102706133807929910935896, 2.40729478207813297329082108844, 2.66074575764918798416278417555, 2.88508412640969975489898991249, 3.07359764668139625014702387533, 3.14788977645547624961585925402, 3.15610927704290823617717056798, 3.31168337844983040532412031550, 3.48573324234880655266363703540, 3.69888710271400459259008466098, 3.91028995471639259639817350450, 4.10356310620724732544164225161, 4.11042014415655920228918629503, 4.22537836108452746765275349877, 4.40466175570534429104058319366, 4.55955623812166782964169344904
Plot not available for L-functions of degree greater than 10.
