| L(s) = 1 | + 2-s − 44·7-s + 9·8-s + 38·11-s − 28·13-s − 44·14-s + 11·16-s + 19·17-s + 187·19-s + 38·22-s + 81·23-s − 28·26-s + 160·29-s + 227·31-s + 58·32-s + 19·34-s − 78·37-s + 187·38-s − 338·41-s − 22·43-s + 81·46-s + 472·47-s + 355·49-s − 521·53-s − 396·56-s + 160·58-s + 140·59-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 2.37·7-s + 0.397·8-s + 1.04·11-s − 0.597·13-s − 0.839·14-s + 0.171·16-s + 0.271·17-s + 2.25·19-s + 0.368·22-s + 0.734·23-s − 0.211·26-s + 1.02·29-s + 1.31·31-s + 0.320·32-s + 0.0958·34-s − 0.346·37-s + 0.798·38-s − 1.28·41-s − 0.0780·43-s + 0.259·46-s + 1.46·47-s + 1.03·49-s − 1.35·53-s − 0.944·56-s + 0.362·58-s + 0.308·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.937169864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.937169864\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} - 5 p T^{3} + p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 44 T + 1581 T^{2} + 31984 T^{3} + 1581 p^{3} T^{4} + 44 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 38 T + 1381 T^{2} - 17876 T^{3} + 1381 p^{3} T^{4} - 38 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 28 T + 155 p T^{2} + 22912 T^{3} + 155 p^{4} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 19 T + 3262 T^{2} + 367193 T^{3} + 3262 p^{3} T^{4} - 19 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 187 T + 24164 T^{2} - 2039395 T^{3} + 24164 p^{3} T^{4} - 187 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 81 T + 13200 T^{2} + 72927 T^{3} + 13200 p^{3} T^{4} - 81 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 160 T + 25399 T^{2} + 88280 T^{3} + 25399 p^{3} T^{4} - 160 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 227 T + 71400 T^{2} - 13771435 T^{3} + 71400 p^{3} T^{4} - 227 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 78 T + 52035 T^{2} - 5735212 T^{3} + 52035 p^{3} T^{4} + 78 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 338 T + 163951 T^{2} + 34473956 T^{3} + 163951 p^{3} T^{4} + 338 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 22 T + 78605 T^{2} - 14966252 T^{3} + 78605 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 472 T + 365677 T^{2} - 97725712 T^{3} + 365677 p^{3} T^{4} - 472 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 521 T + 508018 T^{2} + 154190045 T^{3} + 508018 p^{3} T^{4} + 521 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 140 T + 449689 T^{2} - 23374640 T^{3} + 449689 p^{3} T^{4} - 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 595 T + 678194 T^{2} - 268324783 T^{3} + 678194 p^{3} T^{4} - 595 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 878 T + 978237 T^{2} + 516844828 T^{3} + 978237 p^{3} T^{4} + 878 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 602 T + 490081 T^{2} + 150373964 T^{3} + 490081 p^{3} T^{4} + 602 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 1294 T + 1071143 T^{2} + 602684716 T^{3} + 1071143 p^{3} T^{4} + 1294 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 629 T + 1576176 T^{2} - 622253365 T^{3} + 1576176 p^{3} T^{4} - 629 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1287 T + 1349484 T^{2} - 1125375327 T^{3} + 1349484 p^{3} T^{4} - 1287 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2154 T + 3172479 T^{2} - 3111332052 T^{3} + 3172479 p^{3} T^{4} - 2154 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1392 T + 3281955 T^{2} + 2604477152 T^{3} + 3281955 p^{3} T^{4} + 1392 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.177470279607106059911822418442, −8.664125895959126253895820511530, −8.457948205417596557045396508922, −8.085052980986179924802340272965, −7.54253803159739043167784171038, −7.48513746218247372118296796378, −6.95906847023383582051152817238, −6.91863843122226248528436597804, −6.61654057060561390362463320771, −6.15021115087886772735711633452, −5.98934696679403548067066027498, −5.76890644407892747934772076619, −5.13965622973339855034720923065, −4.91206280431900223880113300253, −4.50492500557719034792850418726, −4.37647788208928086531135069117, −3.54230582576365363053905304867, −3.43242353981598250967734743838, −3.18214741212199235923671137288, −2.93882009687339129219347215655, −2.48000109229321101592908827501, −1.73977946359141147692596194003, −1.28831261844016775940692971992, −0.73835726156009355318343411011, −0.43724613678547454417322215859,
0.43724613678547454417322215859, 0.73835726156009355318343411011, 1.28831261844016775940692971992, 1.73977946359141147692596194003, 2.48000109229321101592908827501, 2.93882009687339129219347215655, 3.18214741212199235923671137288, 3.43242353981598250967734743838, 3.54230582576365363053905304867, 4.37647788208928086531135069117, 4.50492500557719034792850418726, 4.91206280431900223880113300253, 5.13965622973339855034720923065, 5.76890644407892747934772076619, 5.98934696679403548067066027498, 6.15021115087886772735711633452, 6.61654057060561390362463320771, 6.91863843122226248528436597804, 6.95906847023383582051152817238, 7.48513746218247372118296796378, 7.54253803159739043167784171038, 8.085052980986179924802340272965, 8.457948205417596557045396508922, 8.664125895959126253895820511530, 9.177470279607106059911822418442
