| L(s) = 1 | − 2-s + 6·3-s + 4-s − 6·6-s − 8-s + 21·9-s + 6·12-s − 8·13-s + 16-s − 21·18-s − 6·24-s + 25-s + 8·26-s + 56·27-s − 16·31-s − 5·32-s + 21·36-s − 16·37-s − 48·39-s − 4·41-s + 6·48-s + 18·49-s − 50-s − 8·52-s + 24·53-s − 56·54-s + 16·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3.46·3-s + 1/2·4-s − 2.44·6-s − 0.353·8-s + 7·9-s + 1.73·12-s − 2.21·13-s + 1/4·16-s − 4.94·18-s − 1.22·24-s + 1/5·25-s + 1.56·26-s + 10.7·27-s − 2.87·31-s − 0.883·32-s + 7/2·36-s − 2.63·37-s − 7.68·39-s − 0.624·41-s + 0.866·48-s + 18/7·49-s − 0.141·50-s − 1.10·52-s + 3.29·53-s − 7.62·54-s + 2.03·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.478569564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.478569564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | \( ( 1 - T )^{6} \) |
| 5 | \( 1 - T^{2} - 8 T^{3} - p T^{4} + p^{3} T^{6} \) |
| good | 7 | \( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 191 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 4 T + 23 T^{2} + 48 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 2255 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 2 p T^{2} + 1775 T^{4} - 40932 T^{6} + 1775 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 8 T + 89 T^{2} + 432 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 8 T + 3 p T^{2} + 584 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 + 2 T + 23 T^{2} + 220 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 65 T^{2} + 64 T^{3} + 65 p T^{4} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 22367 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 12 T + 191 T^{2} - 1264 T^{3} + 191 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 137 T^{2} - 64 T^{3} + 137 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 8 T + 133 T^{2} - 1008 T^{3} + 133 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 2367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 8 T + 185 T^{2} + 880 T^{3} + 185 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 10 T + 103 T^{2} + 396 T^{3} + 103 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 39183 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−7.60251564606655788141626293867, −7.26016426299637003994358224795, −7.19402210933065850364610019597, −7.14795579140369143785259763848, −7.02996087629922503283251163160, −6.90358609831350359136925436865, −6.89671113950949232112588771456, −6.10440114465324932228124064921, −5.77586378404137465217612326758, −5.64163550357426164207865405387, −5.58983139425301669227575689789, −4.93739493998977673826543560308, −4.90721198771548058073980933131, −4.82655512540684729488444429724, −4.25814660963111476603875536677, −3.89068482881488814454791372069, −3.82509730690766540785535225785, −3.52678672159251582304286585508, −3.46236416396936793779091048081, −3.01815391503721367543156529001, −2.46582214543548465582949599927, −2.38843319424567991974975107363, −2.33994703291733375815870286356, −1.84488632238172924083358514137, −1.44147879188092947875719542607,
1.44147879188092947875719542607, 1.84488632238172924083358514137, 2.33994703291733375815870286356, 2.38843319424567991974975107363, 2.46582214543548465582949599927, 3.01815391503721367543156529001, 3.46236416396936793779091048081, 3.52678672159251582304286585508, 3.82509730690766540785535225785, 3.89068482881488814454791372069, 4.25814660963111476603875536677, 4.82655512540684729488444429724, 4.90721198771548058073980933131, 4.93739493998977673826543560308, 5.58983139425301669227575689789, 5.64163550357426164207865405387, 5.77586378404137465217612326758, 6.10440114465324932228124064921, 6.89671113950949232112588771456, 6.90358609831350359136925436865, 7.02996087629922503283251163160, 7.14795579140369143785259763848, 7.19402210933065850364610019597, 7.26016426299637003994358224795, 7.60251564606655788141626293867
Plot not available for L-functions of degree greater than 10.
