| L(s) = 1 | − 3-s + 3·5-s − 2·7-s + 9-s − 7·11-s − 13-s − 3·15-s + 10·17-s + 13·19-s + 2·21-s − 3·23-s + 6·25-s + 27-s + 13·29-s + 8·31-s + 7·33-s − 6·35-s + 5·37-s + 39-s + 8·41-s − 24·43-s + 3·45-s − 2·47-s − 9·49-s − 10·51-s + 53-s − 21·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.755·7-s + 1/3·9-s − 2.11·11-s − 0.277·13-s − 0.774·15-s + 2.42·17-s + 2.98·19-s + 0.436·21-s − 0.625·23-s + 6/5·25-s + 0.192·27-s + 2.41·29-s + 1.43·31-s + 1.21·33-s − 1.01·35-s + 0.821·37-s + 0.160·39-s + 1.24·41-s − 3.65·43-s + 0.447·45-s − 0.291·47-s − 9/7·49-s − 1.40·51-s + 0.137·53-s − 2.83·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.274835136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.274835136\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T - 2 T^{3} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 27 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 7 T + 40 T^{2} + 140 T^{3} + 40 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 16 T^{2} - 24 T^{3} + 16 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 10 T + 61 T^{2} - 257 T^{3} + 61 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 13 T + 90 T^{2} - 432 T^{3} + 90 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 13 T + 113 T^{2} - 678 T^{3} + 113 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 105 T^{2} - 495 T^{3} + 105 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 5 T + 19 T^{2} + 126 T^{3} + 19 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 121 T^{2} - 655 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 49 T^{2} - 212 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - T + 59 T^{2} - 406 T^{3} + 59 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 17 T + 243 T^{2} - 2042 T^{3} + 243 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 132 T^{2} - 866 T^{3} + 132 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 5 T + 109 T^{2} + 174 T^{3} + 109 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 22 T + 309 T^{2} - 2899 T^{3} + 309 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 43 T^{2} + 368 T^{3} + 43 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 93 T^{2} - 120 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 15 T + 289 T^{2} - 2454 T^{3} + 289 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 139 T^{2} + 1360 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 210 T^{2} + 632 T^{3} + 210 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.219588866434992146123950448801, −7.85334589747077589175542377544, −7.78835024458228606068362767267, −7.49365750999893591382265220268, −7.03127122098062682646717401097, −6.89249263015458572121416746154, −6.44864329636794283897094501435, −6.25119093457424998147179163532, −6.22084020494635736758782841765, −5.64539554487762014243368590645, −5.28124812088267100563230489234, −5.19421547260630377707680400286, −5.12926823210431549953248099681, −5.00024952232133145829992813807, −4.46105389608642678139232982384, −3.90981480904485413376078697561, −3.40060619205840993311204969511, −3.22545358741562915891247383788, −3.03751760170496031134323152463, −2.72278621476486002711094068346, −2.30667044217186402317908011948, −1.95445209653872842638962824825, −1.23239733781161442692016829059, −0.819094369520520870719249001010, −0.73536150366520661901555770619,
0.73536150366520661901555770619, 0.819094369520520870719249001010, 1.23239733781161442692016829059, 1.95445209653872842638962824825, 2.30667044217186402317908011948, 2.72278621476486002711094068346, 3.03751760170496031134323152463, 3.22545358741562915891247383788, 3.40060619205840993311204969511, 3.90981480904485413376078697561, 4.46105389608642678139232982384, 5.00024952232133145829992813807, 5.12926823210431549953248099681, 5.19421547260630377707680400286, 5.28124812088267100563230489234, 5.64539554487762014243368590645, 6.22084020494635736758782841765, 6.25119093457424998147179163532, 6.44864329636794283897094501435, 6.89249263015458572121416746154, 7.03127122098062682646717401097, 7.49365750999893591382265220268, 7.78835024458228606068362767267, 7.85334589747077589175542377544, 8.219588866434992146123950448801
