| L(s) = 1 | − 3-s + 7-s + 9-s − 19-s − 21-s − 25-s + 31-s + 8·37-s + 49-s + 57-s + 7·61-s + 63-s + 3·67-s + 3·73-s + 75-s − 3·79-s − 93-s + 103-s − 109-s − 8·111-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s − 147-s + ⋯ |
| L(s) = 1 | − 3-s + 7-s + 9-s − 19-s − 21-s − 25-s + 31-s + 8·37-s + 49-s + 57-s + 7·61-s + 63-s + 3·67-s + 3·73-s + 75-s − 3·79-s − 93-s + 103-s − 109-s − 8·111-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s − 147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.615951726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.615951726\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 7 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \) |
| good | 5 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 11 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 13 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 17 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 19 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 23 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 37 | \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 43 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 59 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 61 | \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 67 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 79 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 89 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
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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
−2.84453884871822355885456961829, −2.66293103715532086770108652658, −2.62550695556913035409182859164, −2.61773302552473009506142427662, −2.60797069367019901747665552432, −2.59597194773686617918502438155, −2.50844190944891189116738291543, −2.43070019048293251382612220999, −2.41634162783821338818507428477, −2.13410371155600160100396561580, −2.10122339311883474117057028454, −2.07790804173037319638578966718, −2.07320503487220443967902801219, −1.82905185714119282530934221579, −1.76611225388265193728008127396, −1.46655236902268370341217876297, −1.45517705064671732413871003382, −1.28376888281546015808384673873, −1.18790844415912775965690583255, −1.14374791223363037929337884004, −0.932992812627650320666213126638, −0.922406693256429764080668415927, −0.797760276661416708732099727987, −0.70739488757119567377894225129, −0.53751670027556986150188607865,
0.53751670027556986150188607865, 0.70739488757119567377894225129, 0.797760276661416708732099727987, 0.922406693256429764080668415927, 0.932992812627650320666213126638, 1.14374791223363037929337884004, 1.18790844415912775965690583255, 1.28376888281546015808384673873, 1.45517705064671732413871003382, 1.46655236902268370341217876297, 1.76611225388265193728008127396, 1.82905185714119282530934221579, 2.07320503487220443967902801219, 2.07790804173037319638578966718, 2.10122339311883474117057028454, 2.13410371155600160100396561580, 2.41634162783821338818507428477, 2.43070019048293251382612220999, 2.50844190944891189116738291543, 2.59597194773686617918502438155, 2.60797069367019901747665552432, 2.61773302552473009506142427662, 2.62550695556913035409182859164, 2.66293103715532086770108652658, 2.84453884871822355885456961829
Plot not available for L-functions of degree greater than 10.
