| L(s) = 1 | − 2-s + 9·3-s − 10·4-s − 9·6-s + 21·7-s + 6·8-s + 54·9-s − 66·11-s − 90·12-s − 102·13-s − 21·14-s + 31·16-s − 152·17-s − 54·18-s − 138·19-s + 189·21-s + 66·22-s + 180·23-s + 54·24-s + 102·26-s + 270·27-s − 210·28-s + 170·29-s − 366·31-s + 77·32-s − 594·33-s + 152·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.73·3-s − 5/4·4-s − 0.612·6-s + 1.13·7-s + 0.265·8-s + 2·9-s − 1.80·11-s − 2.16·12-s − 2.17·13-s − 0.400·14-s + 0.484·16-s − 2.16·17-s − 0.707·18-s − 1.66·19-s + 1.96·21-s + 0.639·22-s + 1.63·23-s + 0.459·24-s + 0.769·26-s + 1.92·27-s − 1.41·28-s + 1.08·29-s − 2.12·31-s + 0.425·32-s − 3.13·33-s + 0.766·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 11 T^{2} + 15 T^{3} + 11 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 p T + 3717 T^{2} + 169572 T^{3} + 3717 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 102 T + 9675 T^{2} + 476884 T^{3} + 9675 p^{3} T^{4} + 102 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 152 T + 20867 T^{2} + 91872 p T^{3} + 20867 p^{3} T^{4} + 152 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 138 T + 26205 T^{2} + 1963716 T^{3} + 26205 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 180 T + 40221 T^{2} - 4151304 T^{3} + 40221 p^{3} T^{4} - 180 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 170 T + 65051 T^{2} - 6611676 T^{3} + 65051 p^{3} T^{4} - 170 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 366 T + 102201 T^{2} + 18296380 T^{3} + 102201 p^{3} T^{4} + 366 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 252 T + 99399 T^{2} + 17307224 T^{3} + 99399 p^{3} T^{4} + 252 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 206 T + 68783 T^{2} + 10173252 T^{3} + 68783 p^{3} T^{4} + 206 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 108 T + 76905 T^{2} - 30874696 T^{3} + 76905 p^{3} T^{4} - 108 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 24 T + 84141 T^{2} - 25877328 T^{3} + 84141 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 354 T + 295827 T^{2} + 51866076 T^{3} + 295827 p^{3} T^{4} + 354 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 880 T + 706697 T^{2} + 338588064 T^{3} + 706697 p^{3} T^{4} + 880 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 870 T + 582363 T^{2} + 282478148 T^{3} + 582363 p^{3} T^{4} + 870 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 96 T + 555537 T^{2} - 22556608 T^{3} + 555537 p^{3} T^{4} - 96 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 1018 T + 1108049 T^{2} + 595462884 T^{3} + 1108049 p^{3} T^{4} + 1018 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 1554 T + 1505463 T^{2} + 1009418700 T^{3} + 1505463 p^{3} T^{4} + 1554 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1620 T + 1787517 T^{2} + 1338821912 T^{3} + 1787517 p^{3} T^{4} + 1620 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 872 T + 923489 T^{2} - 308826096 T^{3} + 923489 p^{3} T^{4} - 872 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1938 T + 3246255 T^{2} + 2913932892 T^{3} + 3246255 p^{3} T^{4} + 1938 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1878 T + 3431919 T^{2} + 3287491396 T^{3} + 3431919 p^{3} T^{4} + 1878 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.819538475689449912868812482621, −9.064833958088578917830660225652, −9.004826201686115361983477887495, −8.922884284251983448267100926833, −8.538596329369072570974163711803, −8.446088159270843384394928847292, −8.108842653541418360537782258931, −7.46784629330888256863547962136, −7.43881280183750826671142006598, −7.31424172670817652854461279708, −6.95145637496633041518377479049, −6.37259028068059591896769865916, −6.03130845278875056318533881117, −5.34817523082413510195631494031, −4.96304942609724286849268170662, −4.95030693261606385161048046659, −4.46868625084255878310518130411, −4.35043869342863227248239490429, −4.09332475899934473945596602556, −3.11367130845981125724373900165, −3.00607626324545575041072274529, −2.54265641262831329528611021053, −2.32845484118719679829219004550, −1.66986109735010790002247406610, −1.51147059072778020828153608034, 0, 0, 0,
1.51147059072778020828153608034, 1.66986109735010790002247406610, 2.32845484118719679829219004550, 2.54265641262831329528611021053, 3.00607626324545575041072274529, 3.11367130845981125724373900165, 4.09332475899934473945596602556, 4.35043869342863227248239490429, 4.46868625084255878310518130411, 4.95030693261606385161048046659, 4.96304942609724286849268170662, 5.34817523082413510195631494031, 6.03130845278875056318533881117, 6.37259028068059591896769865916, 6.95145637496633041518377479049, 7.31424172670817652854461279708, 7.43881280183750826671142006598, 7.46784629330888256863547962136, 8.108842653541418360537782258931, 8.446088159270843384394928847292, 8.538596329369072570974163711803, 8.922884284251983448267100926833, 9.004826201686115361983477887495, 9.064833958088578917830660225652, 9.819538475689449912868812482621
