| L(s) = 1 | + 3-s − 4-s + 7-s − 3·8-s − 4·9-s + 11·11-s − 12-s + 5·13-s − 16-s + 12·17-s + 12·19-s + 21-s − 12·23-s − 3·24-s − 15·25-s − 6·27-s − 28-s − 2·29-s − 10·31-s + 6·32-s + 11·33-s + 4·36-s + 2·37-s + 5·39-s + 2·41-s + 5·43-s − 11·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 4/3·9-s + 3.31·11-s − 0.288·12-s + 1.38·13-s − 1/4·16-s + 2.91·17-s + 2.75·19-s + 0.218·21-s − 2.50·23-s − 0.612·24-s − 3·25-s − 1.15·27-s − 0.188·28-s − 0.371·29-s − 1.79·31-s + 1.06·32-s + 1.91·33-s + 2/3·36-s + 0.328·37-s + 0.800·39-s + 0.312·41-s + 0.762·43-s − 1.65·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(503^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(503^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.970790817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.970790817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 503 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T^{2} + 3 T^{3} + p T^{4} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - T + 5 T^{2} - p T^{3} + 5 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 7 | $S_4\times C_2$ | \( 1 - T + 13 T^{2} - 11 T^{3} + 13 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - p T + 69 T^{2} - 277 T^{3} + 69 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 37 T^{2} - 105 T^{3} + 37 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 12 T + 79 T^{2} - 368 T^{3} + 79 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 47 T^{2} + 188 T^{3} + 47 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 10 T + 3 p T^{2} + 548 T^{3} + 3 p^{2} T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 43 T^{2} - 204 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 284 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 5 T + 81 T^{2} - 435 T^{3} + 81 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 19 T + 237 T^{2} - 1849 T^{3} + 237 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 127 T^{2} - 836 T^{3} + 127 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 5 T + 161 T^{2} + 591 T^{3} + 161 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 35 T + 589 T^{2} - 6009 T^{3} + 589 p T^{4} - 35 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 241 T^{2} - 1712 T^{3} + 241 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 35 T + 605 T^{2} - 6403 T^{3} + 605 p T^{4} - 35 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 7 T + 85 T^{2} - 39 T^{3} + 85 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + T + 241 T^{2} + 163 T^{3} + 241 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 4 T + 175 T^{2} - 880 T^{3} + 175 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 23 T + 329 T^{2} + 3379 T^{3} + 329 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.659354643139211161475068573701, −9.429197290545393277824965444233, −9.363122693655639067008573650062, −8.965226817018298697378078685596, −8.355564340313463925359138371172, −8.340288154042382636149119925324, −8.022593457580810139579729364699, −7.81195689875417512458495789578, −7.42258696663777023058769626545, −6.89802886609464085121091108866, −6.68075267731393162194784248717, −5.97822306518092201198192349020, −5.91246773657612240997375878283, −5.62561880297439465388462578826, −5.61200463532765327455218268820, −5.18240044335181127201601851061, −4.07755097682401091816323679767, −3.87540961262458159797339684452, −3.80060202536937614821137323602, −3.46295016204733397794952120390, −3.35927650573894485968211194833, −2.30777564194275348621960509992, −2.05669483713524567618885650905, −1.17100951925839039355875305040, −0.903566396325004342303843257014,
0.903566396325004342303843257014, 1.17100951925839039355875305040, 2.05669483713524567618885650905, 2.30777564194275348621960509992, 3.35927650573894485968211194833, 3.46295016204733397794952120390, 3.80060202536937614821137323602, 3.87540961262458159797339684452, 4.07755097682401091816323679767, 5.18240044335181127201601851061, 5.61200463532765327455218268820, 5.62561880297439465388462578826, 5.91246773657612240997375878283, 5.97822306518092201198192349020, 6.68075267731393162194784248717, 6.89802886609464085121091108866, 7.42258696663777023058769626545, 7.81195689875417512458495789578, 8.022593457580810139579729364699, 8.340288154042382636149119925324, 8.355564340313463925359138371172, 8.965226817018298697378078685596, 9.363122693655639067008573650062, 9.429197290545393277824965444233, 9.659354643139211161475068573701
