| L(s) = 1 | + 2·4-s − 7-s − 3·13-s + 16-s + 25-s − 2·28-s − 13·37-s + 43-s + 49-s − 6·52-s + 7·61-s + 2·67-s + 2·79-s + 3·91-s − 3·97-s + 2·100-s − 3·103-s + 109-s − 112-s − 121-s + 127-s + 131-s + 137-s + 139-s − 26·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·4-s − 7-s − 3·13-s + 16-s + 25-s − 2·28-s − 13·37-s + 43-s + 49-s − 6·52-s + 7·61-s + 2·67-s + 2·79-s + 3·91-s − 3·97-s + 2·100-s − 3·103-s + 109-s − 112-s − 121-s + 127-s + 131-s + 137-s + 139-s − 26·148-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.597329617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.597329617\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| good | 2 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 5 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 11 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 13 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 17 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 23 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 29 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 31 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 37 | \( ( 1 + T )^{12}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 53 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 61 | \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 67 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 79 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 83 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 97 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.69560200876943562103179066782, −2.68941510415299302864919279222, −2.61898857511238789623939382753, −2.56739494023270630183016171024, −2.45791272737740348052753047611, −2.31712399606218457416427949611, −2.30424936780328246524353606565, −2.17053124571437284409861594508, −2.06353884306407187581620360760, −1.95297044228519951975089663064, −1.93668820033489476624710323548, −1.89027554782593363911365266694, −1.74969525720602604271333725168, −1.67081107853750593970545667347, −1.59692863632769010228826221830, −1.58206160332773536016319983584, −1.57085427512501838837208577567, −1.46846682725775958676270420625, −1.17210555900211459284678047767, −1.03490585043092686376852561667, −0.789482722549281147672072698049, −0.69088943876490711545441200411, −0.47773207425061621606353911142, −0.46481250494549227139327711678, −0.43963459911706736213690792734,
0.43963459911706736213690792734, 0.46481250494549227139327711678, 0.47773207425061621606353911142, 0.69088943876490711545441200411, 0.789482722549281147672072698049, 1.03490585043092686376852561667, 1.17210555900211459284678047767, 1.46846682725775958676270420625, 1.57085427512501838837208577567, 1.58206160332773536016319983584, 1.59692863632769010228826221830, 1.67081107853750593970545667347, 1.74969525720602604271333725168, 1.89027554782593363911365266694, 1.93668820033489476624710323548, 1.95297044228519951975089663064, 2.06353884306407187581620360760, 2.17053124571437284409861594508, 2.30424936780328246524353606565, 2.31712399606218457416427949611, 2.45791272737740348052753047611, 2.56739494023270630183016171024, 2.61898857511238789623939382753, 2.68941510415299302864919279222, 2.69560200876943562103179066782
Plot not available for L-functions of degree greater than 10.
