| L(s) = 1 | + 4·2-s + 4·4-s − 4·8-s − 3·9-s − 2·13-s − 8·16-s − 12·17-s − 12·18-s + 14·19-s + 19·25-s − 8·26-s − 48·34-s − 12·36-s + 56·38-s + 2·43-s + 28·47-s − 3·49-s + 76·50-s − 8·52-s − 12·53-s + 28·59-s − 4·64-s − 24·67-s − 48·68-s + 12·72-s + 56·76-s + 6·81-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2·4-s − 1.41·8-s − 9-s − 0.554·13-s − 2·16-s − 2.91·17-s − 2.82·18-s + 3.21·19-s + 19/5·25-s − 1.56·26-s − 8.23·34-s − 2·36-s + 9.08·38-s + 0.304·43-s + 4.08·47-s − 3/7·49-s + 10.7·50-s − 1.10·52-s − 1.64·53-s + 3.64·59-s − 1/2·64-s − 2.93·67-s − 5.82·68-s + 1.41·72-s + 6.42·76-s + 2/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.890451661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.890451661\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{3} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| 17 | \( 1 + 12 T + 83 T^{2} + 392 T^{3} + 83 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( ( 1 - p T + p^{2} T^{2} - 3 p T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 - 19 T^{2} + 162 T^{4} - 919 T^{6} + 162 p^{2} T^{8} - 19 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 47 T^{2} + 1038 T^{4} - 14099 T^{6} + 1038 p^{2} T^{8} - 47 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + T + 2 p T^{2} + 3 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 - 7 T + 70 T^{2} - 271 T^{3} + 70 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 71 T^{2} + 2478 T^{4} - 63203 T^{6} + 2478 p^{2} T^{8} - 71 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 162 T^{2} + 11255 T^{4} - 429068 T^{6} + 11255 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( 1 - 158 T^{2} + 11047 T^{4} - 440720 T^{6} + 11047 p^{2} T^{8} - 158 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 154 T^{2} + 10891 T^{4} - 484912 T^{6} + 10891 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 223 T^{2} + 21542 T^{4} - 1154535 T^{6} + 21542 p^{2} T^{8} - 223 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - T + 76 T^{2} - 217 T^{3} + 76 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( ( 1 - 14 T + 145 T^{2} - 1014 T^{3} + 145 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 + 6 T + 137 T^{2} + 514 T^{3} + 137 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( ( 1 - 14 T + 193 T^{2} - 1422 T^{3} + 193 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( 1 - 82 T^{2} + 1775 T^{4} - 60168 T^{6} + 1775 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 12 T + 89 T^{2} + 604 T^{3} + 89 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 - 98 T^{2} + 11727 T^{4} - 845420 T^{6} + 11727 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 138 T^{2} + 18303 T^{4} - 1472524 T^{6} + 18303 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( 1 + 46 T^{2} + 715 T^{4} - 199424 T^{6} + 715 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 16 T + 273 T^{2} - 2640 T^{3} + 273 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 8 T + 203 T^{2} + 1504 T^{3} + 203 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 250 T^{2} + 43535 T^{4} - 4887660 T^{6} + 43535 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.94198037855641249373282410753, −5.84528103235900689940143389184, −5.65372129823184107658826654602, −5.60295335655990068079615962212, −5.41503879342149395752606968810, −5.22607051775167748358954933808, −5.09582969945548044977539813874, −4.75441413740991324765730204411, −4.62507215911845802263415741882, −4.53311752144200065891665574188, −4.46744778039192477458673128981, −4.36768887542244328533530474927, −4.26399080512577102550800242519, −3.58215024939891053160806191324, −3.51853465519705374927020909830, −3.43226185131747918955277691799, −3.22529083826253773089832408657, −3.13843481398489898228176580190, −2.60927948028672519806040595415, −2.46596390334713804129975898683, −2.28261966680719741529164725766, −2.04241830571489699337641536563, −1.30623489778382231548444434294, −0.879783169499310090275501419239, −0.65330085683607734884152749976,
0.65330085683607734884152749976, 0.879783169499310090275501419239, 1.30623489778382231548444434294, 2.04241830571489699337641536563, 2.28261966680719741529164725766, 2.46596390334713804129975898683, 2.60927948028672519806040595415, 3.13843481398489898228176580190, 3.22529083826253773089832408657, 3.43226185131747918955277691799, 3.51853465519705374927020909830, 3.58215024939891053160806191324, 4.26399080512577102550800242519, 4.36768887542244328533530474927, 4.46744778039192477458673128981, 4.53311752144200065891665574188, 4.62507215911845802263415741882, 4.75441413740991324765730204411, 5.09582969945548044977539813874, 5.22607051775167748358954933808, 5.41503879342149395752606968810, 5.60295335655990068079615962212, 5.65372129823184107658826654602, 5.84528103235900689940143389184, 5.94198037855641249373282410753
Plot not available for L-functions of degree greater than 10.
