| L(s) = 1 | + 6·3-s − 6·7-s − 8-s + 21·9-s − 6·11-s − 6·13-s − 12·17-s − 36·21-s + 6·23-s − 6·24-s + 55·27-s + 6·31-s − 36·33-s − 12·37-s − 36·39-s + 3·41-s − 6·43-s − 6·47-s + 21·49-s − 72·51-s − 30·53-s + 6·56-s + 15·59-s + 6·61-s − 126·63-s + 18·67-s + 36·69-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 2.26·7-s − 0.353·8-s + 7·9-s − 1.80·11-s − 1.66·13-s − 2.91·17-s − 7.85·21-s + 1.25·23-s − 1.22·24-s + 10.5·27-s + 1.07·31-s − 6.26·33-s − 1.97·37-s − 5.76·39-s + 0.468·41-s − 0.914·43-s − 0.875·47-s + 3·49-s − 10.0·51-s − 4.12·53-s + 0.801·56-s + 1.95·59-s + 0.768·61-s − 15.8·63-s + 2.19·67-s + 4.33·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.644524128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.644524128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{3} + T^{6} \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2 p T + 5 p T^{2} - 19 T^{3} + p T^{4} + 17 p T^{5} - 134 T^{6} + 17 p^{2} T^{7} + p^{3} T^{8} - 19 p^{3} T^{9} + 5 p^{5} T^{10} - 2 p^{6} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 - 22 T^{3} + 359 T^{6} - 22 p^{3} T^{9} + p^{6} T^{12} \) |
| 7 | \( 1 + 6 T + 15 T^{2} + 6 T^{3} - 66 T^{4} - 30 p T^{5} - 565 T^{6} - 30 p^{2} T^{7} - 66 p^{2} T^{8} + 6 p^{3} T^{9} + 15 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 6 T - 10 T^{3} + 222 T^{4} + 42 T^{5} - 3181 T^{6} + 42 p T^{7} + 222 p^{2} T^{8} - 10 p^{3} T^{9} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 6 T + 12 T^{2} + 14 T^{3} - 216 T^{4} - 1188 T^{5} - 2757 T^{6} - 1188 p T^{7} - 216 p^{2} T^{8} + 14 p^{3} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + 12 T + 72 T^{2} + 270 T^{3} + 972 T^{4} + 4980 T^{5} + 23779 T^{6} + 4980 p T^{7} + 972 p^{2} T^{8} + 270 p^{3} T^{9} + 72 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 6 T + 24 T^{2} - 64 T^{3} - 84 T^{4} + 4500 T^{5} - 24523 T^{6} + 4500 p T^{7} - 84 p^{2} T^{8} - 64 p^{3} T^{9} + 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 36 T^{2} - 68 T^{3} - 252 T^{4} + 1818 T^{5} + 28295 T^{6} + 1818 p T^{7} - 252 p^{2} T^{8} - 68 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( 1 - 6 T - 33 T^{2} + 346 T^{3} + 342 T^{4} - 6318 T^{5} + 21795 T^{6} - 6318 p T^{7} + 342 p^{2} T^{8} + 346 p^{3} T^{9} - 33 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 6 T + 87 T^{2} + 308 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 3 T + 6 T^{2} - 8 T^{3} + 249 T^{4} - 5409 T^{5} - 39031 T^{6} - 5409 p T^{7} + 249 p^{2} T^{8} - 8 p^{3} T^{9} + 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 6 T - 12 T^{2} - 440 T^{3} - 2358 T^{4} + 11970 T^{5} + 159561 T^{6} + 11970 p T^{7} - 2358 p^{2} T^{8} - 440 p^{3} T^{9} - 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 + 6 T + 48 T^{2} + 290 T^{3} + 588 T^{4} + 13266 T^{5} + 77981 T^{6} + 13266 p T^{7} + 588 p^{2} T^{8} + 290 p^{3} T^{9} + 48 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 30 T + 384 T^{2} + 2384 T^{3} - 2028 T^{4} - 205344 T^{5} - 2162401 T^{6} - 205344 p T^{7} - 2028 p^{2} T^{8} + 2384 p^{3} T^{9} + 384 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 15 T + 72 T^{2} + 270 T^{3} - 4185 T^{4} + 12009 T^{5} - 17927 T^{6} + 12009 p T^{7} - 4185 p^{2} T^{8} + 270 p^{3} T^{9} + 72 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 6 T + 96 T^{2} - 1234 T^{3} + 11628 T^{4} - 85302 T^{5} + 953415 T^{6} - 85302 p T^{7} + 11628 p^{2} T^{8} - 1234 p^{3} T^{9} + 96 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 18 T + 81 T^{2} + 1125 T^{3} - 13365 T^{4} - 1881 T^{5} + 747782 T^{6} - 1881 p T^{7} - 13365 p^{2} T^{8} + 1125 p^{3} T^{9} + 81 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 18 T + 144 T^{2} - 76 T^{3} - 4608 T^{4} + 58986 T^{5} - 408079 T^{6} + 58986 p T^{7} - 4608 p^{2} T^{8} - 76 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 33 T + 486 T^{2} - 3660 T^{3} + 1593 T^{4} + 296733 T^{5} - 3685879 T^{6} + 296733 p T^{7} + 1593 p^{2} T^{8} - 3660 p^{3} T^{9} + 486 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 12 T + 96 T^{2} + 1646 T^{3} + 11484 T^{4} + 81954 T^{5} + 1197729 T^{6} + 81954 p T^{7} + 11484 p^{2} T^{8} + 1646 p^{3} T^{9} + 96 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 6 T - 186 T^{2} - 558 T^{3} + 25188 T^{4} + 30012 T^{5} - 2301977 T^{6} + 30012 p T^{7} + 25188 p^{2} T^{8} - 558 p^{3} T^{9} - 186 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 36 T + 684 T^{2} - 9558 T^{3} + 117144 T^{4} - 1282392 T^{5} + 12697723 T^{6} - 1282392 p T^{7} + 117144 p^{2} T^{8} - 9558 p^{3} T^{9} + 684 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 6 T + 15 T^{2} + 19 T^{3} - 279 T^{4} - 29097 T^{5} - 998226 T^{6} - 29097 p T^{7} - 279 p^{2} T^{8} + 19 p^{3} T^{9} + 15 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.47550685527546180752164818441, −5.34309119980350395943933164083, −5.04665637047388816618815970591, −4.94323859567521483493931398414, −4.71837015058916641015303025900, −4.68150884345896726489905263866, −4.59130075219019827228426576860, −4.37497030464239569246399901584, −4.03151363832239569109410712648, −3.80000160683107462266079690380, −3.52375857245775202365666341109, −3.48438479045925389818145268374, −3.36535679394690600630186610188, −3.29147578667384449464571339826, −3.02675647567413371714970919606, −2.84160017052854163810329503593, −2.57175993283454399852336348925, −2.42440842108811029778468413040, −2.20093801501891534116050946743, −2.08933183538911546845797259667, −2.07860501360803805221682047096, −1.84752818571102867198932567584, −1.05951074331319831474131295821, −0.76495207097521342370367671049, −0.17872596496707226183241670003,
0.17872596496707226183241670003, 0.76495207097521342370367671049, 1.05951074331319831474131295821, 1.84752818571102867198932567584, 2.07860501360803805221682047096, 2.08933183538911546845797259667, 2.20093801501891534116050946743, 2.42440842108811029778468413040, 2.57175993283454399852336348925, 2.84160017052854163810329503593, 3.02675647567413371714970919606, 3.29147578667384449464571339826, 3.36535679394690600630186610188, 3.48438479045925389818145268374, 3.52375857245775202365666341109, 3.80000160683107462266079690380, 4.03151363832239569109410712648, 4.37497030464239569246399901584, 4.59130075219019827228426576860, 4.68150884345896726489905263866, 4.71837015058916641015303025900, 4.94323859567521483493931398414, 5.04665637047388816618815970591, 5.34309119980350395943933164083, 5.47550685527546180752164818441
Plot not available for L-functions of degree greater than 10.
