| L(s) = 1 | − 2-s + 4·3-s + 4-s − 8·5-s − 4·6-s − 22·7-s + 15·8-s − 3·9-s + 8·10-s + 4·12-s − 30·13-s + 22·14-s − 32·15-s − 31·16-s + 51·17-s + 3·18-s − 80·19-s − 8·20-s − 88·21-s + 142·23-s + 60·24-s − 267·25-s + 30·26-s − 76·27-s − 22·28-s + 456·29-s + 32·30-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.769·3-s + 1/8·4-s − 0.715·5-s − 0.272·6-s − 1.18·7-s + 0.662·8-s − 1/9·9-s + 0.252·10-s + 0.0962·12-s − 0.640·13-s + 0.419·14-s − 0.550·15-s − 0.484·16-s + 0.727·17-s + 0.0392·18-s − 0.965·19-s − 0.0894·20-s − 0.914·21-s + 1.28·23-s + 0.510·24-s − 2.13·25-s + 0.226·26-s − 0.541·27-s − 0.148·28-s + 2.91·29-s + 0.194·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.006931048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.006931048\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T - p^{4} T^{3} + p^{6} T^{5} + p^{9} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 4 T + 19 T^{2} - 4 p T^{3} + 19 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 8 T + 331 T^{2} + 2032 T^{3} + 331 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 22 T + 891 T^{2} + 14300 T^{3} + 891 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 30 T + 5119 T^{2} + 141212 T^{3} + 5119 p^{3} T^{4} + 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 80 T + 15945 T^{2} + 757312 T^{3} + 15945 p^{3} T^{4} + 80 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 142 T + 20731 T^{2} - 1854884 T^{3} + 20731 p^{3} T^{4} - 142 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 456 T + 127075 T^{2} - 23761392 T^{3} + 127075 p^{3} T^{4} - 456 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 230 T + 77787 T^{2} - 13785468 T^{3} + 77787 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 356 T + 133995 T^{2} - 29888184 T^{3} + 133995 p^{3} T^{4} - 356 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 294 T + 120199 T^{2} - 38886804 T^{3} + 120199 p^{3} T^{4} - 294 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 556 T + 289617 T^{2} + 81141512 T^{3} + 289617 p^{3} T^{4} + 556 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 640 T + 396797 T^{2} - 134564608 T^{3} + 396797 p^{3} T^{4} - 640 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 302 T + 293171 T^{2} - 71759636 T^{3} + 293171 p^{3} T^{4} - 302 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 636 T + 514369 T^{2} - 211823016 T^{3} + 514369 p^{3} T^{4} - 636 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 84 T + 556531 T^{2} - 31340024 T^{3} + 556531 p^{3} T^{4} - 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1008 T + 967329 T^{2} - 607104160 T^{3} + 967329 p^{3} T^{4} - 1008 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 402 T + 483859 T^{2} + 12894428 T^{3} + 483859 p^{3} T^{4} + 402 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 838 T + 1394903 T^{2} + 671950004 T^{3} + 1394903 p^{3} T^{4} + 838 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 594 T + 357843 T^{2} + 156405492 T^{3} + 357843 p^{3} T^{4} - 594 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2396 T + 3204249 T^{2} - 2882084008 T^{3} + 3204249 p^{3} T^{4} - 2396 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 170 T + 1042603 T^{2} - 206881916 T^{3} + 1042603 p^{3} T^{4} + 170 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 270 T + 2151919 T^{2} + 286220420 T^{3} + 2151919 p^{3} T^{4} + 270 p^{6} T^{5} + p^{9} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.71700051337580701928695653788, −7.64488921802742296125998815680, −7.20819954365180187454281683550, −7.17367887823359239288694964299, −6.56329104278644750513670524391, −6.44836621457614135073639531948, −6.31228913819262299891845537188, −6.10437714491567237073831807789, −5.50523700706601678118325912934, −5.37424587543280058956334443170, −4.83504255780244676415191012547, −4.66377844169000773034143964439, −4.53217521302348666620239906620, −3.91962195620552579984493751610, −3.79222166024328542333127754666, −3.65193343685797758446714098227, −3.12166306178641343522814343994, −2.67448162188042488907448079496, −2.64279012846054790938562446091, −2.42830401546173593239456516794, −1.88913031476043498957846400583, −1.39734133794656896063154665323, −0.966599422566915523265720247724, −0.49621404286406076158031233161, −0.43521073710083913612191019361,
0.43521073710083913612191019361, 0.49621404286406076158031233161, 0.966599422566915523265720247724, 1.39734133794656896063154665323, 1.88913031476043498957846400583, 2.42830401546173593239456516794, 2.64279012846054790938562446091, 2.67448162188042488907448079496, 3.12166306178641343522814343994, 3.65193343685797758446714098227, 3.79222166024328542333127754666, 3.91962195620552579984493751610, 4.53217521302348666620239906620, 4.66377844169000773034143964439, 4.83504255780244676415191012547, 5.37424587543280058956334443170, 5.50523700706601678118325912934, 6.10437714491567237073831807789, 6.31228913819262299891845537188, 6.44836621457614135073639531948, 6.56329104278644750513670524391, 7.17367887823359239288694964299, 7.20819954365180187454281683550, 7.64488921802742296125998815680, 7.71700051337580701928695653788
