L(s) = 1 | − 4-s + 16-s + 25-s + 5·37-s + 4·43-s − 7·61-s + 2·67-s + 2·79-s − 100-s − 2·109-s − 121-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | − 4-s + 16-s + 25-s + 5·37-s + 4·43-s − 7·61-s + 2·67-s + 2·79-s − 100-s − 2·109-s − 121-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.976176397\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.976176397\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 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 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 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 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 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 | \( ( 1 - T + T^{2} )^{6}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4} \) |
| 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 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 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 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 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 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
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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
−2.77153095546622395154564629368, −2.71162264983965841767537907929, −2.70868227105648732351642339655, −2.68600580678380337828137333889, −2.60667052511229924149748231322, −2.53245077371757968108067556326, −2.48348547066613168651994531683, −2.37442514326450320999576569180, −2.29291799613894351956057376218, −1.96719233747450810223289956386, −1.96309414752744157661307984109, −1.92371446902964919450294953681, −1.80147643872881858122417135592, −1.78473389363134004270693515461, −1.63768802767827839644562157770, −1.35476252420495520108719072637, −1.33052893230546012538279255029, −1.25255059048672975979773948787, −1.19020283217464445093368029592, −1.11891633270732890297041529812, −1.07884469038645773374894386451, −0.800793401987249327787789938159, −0.71834366968979137209035273473, −0.51126986263210811498919477005, −0.38855456597348326962304894924,
0.38855456597348326962304894924, 0.51126986263210811498919477005, 0.71834366968979137209035273473, 0.800793401987249327787789938159, 1.07884469038645773374894386451, 1.11891633270732890297041529812, 1.19020283217464445093368029592, 1.25255059048672975979773948787, 1.33052893230546012538279255029, 1.35476252420495520108719072637, 1.63768802767827839644562157770, 1.78473389363134004270693515461, 1.80147643872881858122417135592, 1.92371446902964919450294953681, 1.96309414752744157661307984109, 1.96719233747450810223289956386, 2.29291799613894351956057376218, 2.37442514326450320999576569180, 2.48348547066613168651994531683, 2.53245077371757968108067556326, 2.60667052511229924149748231322, 2.68600580678380337828137333889, 2.70868227105648732351642339655, 2.71162264983965841767537907929, 2.77153095546622395154564629368
Plot not available for L-functions of degree greater than 10.