L(s) = 1 | + 3·9-s − 6·41-s + 6·49-s − 12·73-s + 3·81-s − 6·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | + 3·9-s − 6·41-s + 6·49-s − 12·73-s + 3·81-s − 6·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4859253585\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4859253585\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T^{6} + T^{12} \) |
good | 3 | \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \) |
| 5 | \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \) |
| 7 | \( ( 1 - T^{2} + T^{4} )^{6} \) |
| 11 | \( ( 1 - T^{6} + T^{12} )^{2} \) |
| 13 | \( ( 1 - T^{6} + T^{12} )^{2} \) |
| 17 | \( ( 1 + T^{3} + T^{6} )^{4} \) |
| 23 | \( ( 1 - T^{6} + T^{12} )^{2} \) |
| 29 | \( ( 1 - T^{6} + T^{12} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \) |
| 37 | \( ( 1 + T^{2} )^{12} \) |
| 41 | \( ( 1 + T + T^{2} )^{6}( 1 - T^{3} + T^{6} )^{2} \) |
| 43 | \( ( 1 - T^{6} + T^{12} )^{2} \) |
| 47 | \( ( 1 - T^{6} + T^{12} )^{2} \) |
| 53 | \( ( 1 - T^{6} + T^{12} )^{2} \) |
| 59 | \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \) |
| 61 | \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \) |
| 67 | \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \) |
| 71 | \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \) |
| 73 | \( ( 1 + T )^{12}( 1 - T^{3} + T^{6} )^{2} \) |
| 79 | \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \) |
| 83 | \( ( 1 - T^{6} + T^{12} )^{2} \) |
| 89 | \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \) |
| 97 | \( ( 1 + T + T^{2} )^{6}( 1 - T^{3} + T^{6} )^{2} \) |
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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
−3.16764822558467111678555864723, −3.13037903922762753027582431588, −3.12674812637773752249467569328, −3.06482814512033183405177947461, −3.00960700019456537680335124638, −2.92694624277638073619558766126, −2.92200061633886908781910854403, −2.55060685991914191454508305256, −2.47497633780502933382292621077, −2.44055951308115548738215604526, −2.42113612990466492892477293578, −2.41509851765868934369956338044, −2.25824319032103595698018379715, −2.12104111770290365155101052704, −1.71730106099862639184364120584, −1.69335723270342150790973741657, −1.62152191060400622263789908734, −1.55430523034076738506449401481, −1.52592928334195601004677255390, −1.40818187545204399092170875035, −1.29478865325345599634721666383, −1.23910926268359474326035321563, −1.13939779746055201226335974365, −0.874569412338624668585024776104, −0.31488439567298742981317308735,
0.31488439567298742981317308735, 0.874569412338624668585024776104, 1.13939779746055201226335974365, 1.23910926268359474326035321563, 1.29478865325345599634721666383, 1.40818187545204399092170875035, 1.52592928334195601004677255390, 1.55430523034076738506449401481, 1.62152191060400622263789908734, 1.69335723270342150790973741657, 1.71730106099862639184364120584, 2.12104111770290365155101052704, 2.25824319032103595698018379715, 2.41509851765868934369956338044, 2.42113612990466492892477293578, 2.44055951308115548738215604526, 2.47497633780502933382292621077, 2.55060685991914191454508305256, 2.92200061633886908781910854403, 2.92694624277638073619558766126, 3.00960700019456537680335124638, 3.06482814512033183405177947461, 3.12674812637773752249467569328, 3.13037903922762753027582431588, 3.16764822558467111678555864723
Plot not available for L-functions of degree greater than 10.