| L(s) = 1 | + 29·4-s − 27·9-s − 66·11-s + 398·16-s + 560·19-s + 580·29-s − 784·31-s − 783·36-s − 1.32e3·41-s − 1.91e3·44-s + 298·49-s + 1.22e3·59-s − 1.16e3·61-s + 3.07e3·64-s − 3.23e3·71-s + 1.62e4·76-s − 248·79-s + 486·81-s − 1.67e3·89-s + 1.78e3·99-s − 2.05e3·101-s + 716·109-s + 1.68e4·116-s + 2.54e3·121-s − 2.27e4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 29/8·4-s − 9-s − 1.80·11-s + 6.21·16-s + 6.76·19-s + 3.71·29-s − 4.54·31-s − 3.62·36-s − 5.04·41-s − 6.55·44-s + 0.868·49-s + 2.70·59-s − 2.44·61-s + 5.99·64-s − 5.40·71-s + 24.5·76-s − 0.353·79-s + 2/3·81-s − 1.99·89-s + 1.80·99-s − 2.02·101-s + 0.629·109-s + 13.4·116-s + 1.90·121-s − 16.4·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(16.52625291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.52625291\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 11 | \( ( 1 + p T )^{6} \) |
| good | 2 | \( 1 - 29 T^{2} + 443 T^{4} - 4375 T^{6} + 443 p^{6} T^{8} - 29 p^{12} T^{10} + p^{18} T^{12} \) |
| 7 | \( 1 - 298 T^{2} + 101375 T^{4} - 46398796 T^{6} + 101375 p^{6} T^{8} - 298 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 - 74 p T^{2} + 3887703 T^{4} - 10121628156 T^{6} + 3887703 p^{6} T^{8} - 74 p^{13} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 19002 T^{2} + 190081135 T^{4} - 1155524897132 T^{6} + 190081135 p^{6} T^{8} - 19002 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 - 280 T + 39201 T^{2} - 3743984 T^{3} + 39201 p^{3} T^{4} - 280 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 26410 T^{2} + 672063967 T^{4} - 8464894539980 T^{6} + 672063967 p^{6} T^{8} - 26410 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 - 10 p T + 46667 T^{2} - 4894124 T^{3} + 46667 p^{3} T^{4} - 10 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( ( 1 + 392 T + 121533 T^{2} + 23039344 T^{3} + 121533 p^{3} T^{4} + 392 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 100898 T^{2} + 2250462855 T^{4} + 30925274913156 T^{6} + 2250462855 p^{6} T^{8} - 100898 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 662 T + 150855 T^{2} + 22689620 T^{3} + 150855 p^{3} T^{4} + 662 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 17666 T^{2} + 3212644199 T^{4} - 872810853291068 T^{6} + 3212644199 p^{6} T^{8} - 17666 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 - 224090 T^{2} + 15678901295 T^{4} - 463473771795884 T^{6} + 15678901295 p^{6} T^{8} - 224090 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 - 768802 T^{2} + 262142669095 T^{4} - 50567765970385916 T^{6} + 262142669095 p^{6} T^{8} - 768802 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 - 612 T + 191353 T^{2} - 89255576 T^{3} + 191353 p^{3} T^{4} - 612 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( ( 1 + 582 T + 700987 T^{2} + 242850884 T^{3} + 700987 p^{3} T^{4} + 582 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 - 142002 T^{2} + 250067943927 T^{4} - 23516035098600220 T^{6} + 250067943927 p^{6} T^{8} - 142002 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 1616 T + 1744933 T^{2} + 1196967136 T^{3} + 1744933 p^{3} T^{4} + 1616 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 2127194 T^{2} + 1958786592351 T^{4} - 997766112220189932 T^{6} + 1958786592351 p^{6} T^{8} - 2127194 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 + 124 T + 1293245 T^{2} + 95402344 T^{3} + 1293245 p^{3} T^{4} + 124 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 2499218 T^{2} + 2898138245143 T^{4} - 2053680401124231004 T^{6} + 2898138245143 p^{6} T^{8} - 2499218 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 838 T + 1110375 T^{2} + 349581812 T^{3} + 1110375 p^{3} T^{4} + 838 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 875258 T^{2} + 1138077455055 T^{4} - 1285624950058536684 T^{6} + 1138077455055 p^{6} T^{8} - 875258 p^{12} T^{10} + p^{18} T^{12} \) |
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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
−5.17084284230164884074195361234, −5.15128348543503557424487829908, −4.79765703751663101373928634863, −4.67176741746502032444641578450, −4.23002191741753153766090178095, −4.22714843025704698540806292845, −3.67848588165221376444947054006, −3.60453612947289667477787866720, −3.42229716384985030690189315023, −3.38742940117793666044599525656, −2.94560931301175477165257764443, −2.94063814150439042296339638387, −2.91107282603252734793412222299, −2.69714838672851145114810526859, −2.65784172380986433086446510095, −2.61914039559904875412110505461, −1.87262720620055642839343324090, −1.71751595518205255511682060602, −1.70299599475119007294995153869, −1.49465740158016522337668973524, −1.48015302299639405485927326152, −1.11215058160818887827375167001, −0.58343602511185087734413377650, −0.56850466959310372496269569540, −0.22036678233296368015150158988,
0.22036678233296368015150158988, 0.56850466959310372496269569540, 0.58343602511185087734413377650, 1.11215058160818887827375167001, 1.48015302299639405485927326152, 1.49465740158016522337668973524, 1.70299599475119007294995153869, 1.71751595518205255511682060602, 1.87262720620055642839343324090, 2.61914039559904875412110505461, 2.65784172380986433086446510095, 2.69714838672851145114810526859, 2.91107282603252734793412222299, 2.94063814150439042296339638387, 2.94560931301175477165257764443, 3.38742940117793666044599525656, 3.42229716384985030690189315023, 3.60453612947289667477787866720, 3.67848588165221376444947054006, 4.22714843025704698540806292845, 4.23002191741753153766090178095, 4.67176741746502032444641578450, 4.79765703751663101373928634863, 5.15128348543503557424487829908, 5.17084284230164884074195361234
Plot not available for L-functions of degree greater than 10.
