| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s − 2·5-s − 9·6-s + 4·7-s − 10·8-s + 4·9-s + 6·10-s − 3·11-s + 18·12-s − 12·14-s − 6·15-s + 15·16-s + 5·17-s − 12·18-s − 19-s − 12·20-s + 12·21-s + 9·22-s − 30·24-s − 3·25-s + 2·27-s + 24·28-s − 10·29-s + 18·30-s + 16·31-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s − 0.894·5-s − 3.67·6-s + 1.51·7-s − 3.53·8-s + 4/3·9-s + 1.89·10-s − 0.904·11-s + 5.19·12-s − 3.20·14-s − 1.54·15-s + 15/4·16-s + 1.21·17-s − 2.82·18-s − 0.229·19-s − 2.68·20-s + 2.61·21-s + 1.91·22-s − 6.12·24-s − 3/5·25-s + 0.384·27-s + 4.53·28-s − 1.85·29-s + 3.28·30-s + 2.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38614472 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38614472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.146229331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146229331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 - p T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 12 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 48 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 29 T^{2} + 53 T^{3} + 29 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 5 T + 29 T^{2} - 73 T^{3} + 29 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + T + 41 T^{2} + 51 T^{3} + 41 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 41 T^{2} + 56 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 10 T + 83 T^{2} + 476 T^{3} + 83 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 16 T + 169 T^{2} - 1096 T^{3} + 169 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 14 T + 167 T^{2} - 1092 T^{3} + 167 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 7 T + 109 T^{2} + 483 T^{3} + 109 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 11 T + 125 T^{2} - 945 T^{3} + 125 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 2 T + 105 T^{2} - 180 T^{3} + 105 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 131 T^{2} - 56 T^{3} + 131 p T^{4} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 117 T^{2} - 481 T^{3} + 117 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 4 T + 39 T^{2} + 424 T^{3} + 39 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 21 T + 285 T^{2} + 2527 T^{3} + 285 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 12 T + 233 T^{2} - 1712 T^{3} + 233 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + T + 133 T^{2} + 397 T^{3} + 133 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 261 T^{2} + 2612 T^{3} + 261 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - T + 233 T^{2} - 179 T^{3} + 233 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 25 T + 361 T^{2} + 3693 T^{3} + 361 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 23 T + 353 T^{2} + 3579 T^{3} + 353 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−10.21169053678005845976424476082, −9.950693021206699968071789598736, −9.557204197765491604035751881303, −9.271315054689697286832554069889, −9.124183720497808064025075784833, −8.448388043929398028587888871291, −8.433025627231612324991871870798, −8.032097583245981141471144508013, −7.928290680994143610446068355397, −7.81517310197648222723600190525, −7.52874605587915491766757492111, −6.96750954989767630616038946645, −6.89531212031146585363038881931, −6.03626399448508789783736714976, −5.65435300109182758421939538692, −5.62703942190026838327956302047, −4.66930375373960719358478139000, −4.27245566155578154403308539282, −4.12473728616083118300334546827, −3.19324996294494397930569399052, −2.88254502174365047261052599536, −2.75664713130300424803391274565, −1.86995629665135213307745984822, −1.68280096950707187455959748741, −0.75336314361189250095040983026,
0.75336314361189250095040983026, 1.68280096950707187455959748741, 1.86995629665135213307745984822, 2.75664713130300424803391274565, 2.88254502174365047261052599536, 3.19324996294494397930569399052, 4.12473728616083118300334546827, 4.27245566155578154403308539282, 4.66930375373960719358478139000, 5.62703942190026838327956302047, 5.65435300109182758421939538692, 6.03626399448508789783736714976, 6.89531212031146585363038881931, 6.96750954989767630616038946645, 7.52874605587915491766757492111, 7.81517310197648222723600190525, 7.928290680994143610446068355397, 8.032097583245981141471144508013, 8.433025627231612324991871870798, 8.448388043929398028587888871291, 9.124183720497808064025075784833, 9.271315054689697286832554069889, 9.557204197765491604035751881303, 9.950693021206699968071789598736, 10.21169053678005845976424476082
