| L(s) = 1 | + 2-s − 3·3-s + 3·5-s − 3·6-s − 4·7-s − 8-s + 6·9-s + 3·10-s + 9·11-s + 6·13-s − 4·14-s − 9·15-s − 3·16-s + 2·17-s + 6·18-s − 4·19-s + 12·21-s + 9·22-s + 23-s + 3·24-s + 6·25-s + 6·26-s − 10·27-s − 3·29-s − 9·30-s − 27·33-s + 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1.34·5-s − 1.22·6-s − 1.51·7-s − 0.353·8-s + 2·9-s + 0.948·10-s + 2.71·11-s + 1.66·13-s − 1.06·14-s − 2.32·15-s − 3/4·16-s + 0.485·17-s + 1.41·18-s − 0.917·19-s + 2.61·21-s + 1.91·22-s + 0.208·23-s + 0.612·24-s + 6/5·25-s + 1.17·26-s − 1.92·27-s − 0.557·29-s − 1.64·30-s − 4.70·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82312875 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82312875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.178221363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178221363\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 2 p T^{2} + 6 p T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 38 T^{2} - 154 T^{3} + 38 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 40 T^{2} - 60 T^{3} + 40 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 64 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - T + 13 T^{2} + 66 T^{3} + 13 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 706 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 13 T + 167 T^{2} - 1094 T^{3} + 167 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 13 T + 131 T^{2} + 810 T^{3} + 131 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 72 T^{2} + 78 T^{3} + 72 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 3 T + 45 T^{2} + 634 T^{3} + 45 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 22 T + 317 T^{2} - 2852 T^{3} + 317 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 187 T^{2} - 1108 T^{3} + 187 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 28 T + 406 T^{2} + 3946 T^{3} + 406 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 488 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 3 T + 35 T^{2} + 10 p T^{3} + 35 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 97 T^{2} + 92 T^{3} + 97 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 15 T + 311 T^{2} + 2534 T^{3} + 311 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 30 T + 450 T^{2} - 4738 T^{3} + 450 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + T + 149 T^{2} + 270 T^{3} + 149 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.11390393472597674177505102814, −9.541472433355950975720708996681, −9.477996869809564280760342564614, −9.178707640873895974168698530948, −8.994230074726734708528024608257, −8.480133861513070054844477896308, −8.288660575807392053528597178994, −7.35638965255827599185218087612, −7.35637204840578365156498073296, −6.60915541576153082151039654840, −6.53819267773425192181825712764, −6.38548047496945300773667973772, −6.10972133769248522691221431287, −6.00338698799123947392620442998, −5.47775641669169881350036698096, −5.29995804282330969784902939486, −4.53167440669236326129400406824, −4.19943067399341164727552304974, −4.18547734109449474598229219907, −3.53323878764179843659283247441, −3.29437045818504435624558241890, −2.56603838914060313534104374174, −1.86006677722456255803498369569, −1.27925173991626807871074967574, −0.808029451296697548656237011504,
0.808029451296697548656237011504, 1.27925173991626807871074967574, 1.86006677722456255803498369569, 2.56603838914060313534104374174, 3.29437045818504435624558241890, 3.53323878764179843659283247441, 4.18547734109449474598229219907, 4.19943067399341164727552304974, 4.53167440669236326129400406824, 5.29995804282330969784902939486, 5.47775641669169881350036698096, 6.00338698799123947392620442998, 6.10972133769248522691221431287, 6.38548047496945300773667973772, 6.53819267773425192181825712764, 6.60915541576153082151039654840, 7.35637204840578365156498073296, 7.35638965255827599185218087612, 8.288660575807392053528597178994, 8.480133861513070054844477896308, 8.994230074726734708528024608257, 9.178707640873895974168698530948, 9.477996869809564280760342564614, 9.541472433355950975720708996681, 10.11390393472597674177505102814
