| L(s) = 1 | − 3·3-s + 3·7-s + 6·9-s + 6·19-s − 9·21-s − 10·27-s − 12·29-s + 3·31-s + 6·37-s − 21·43-s − 12·47-s + 6·49-s − 6·53-s − 18·57-s + 6·59-s + 3·61-s + 18·63-s + 27·67-s + 12·71-s − 9·73-s + 9·79-s + 15·81-s − 18·83-s + 36·87-s + 24·89-s − 9·93-s + 21·97-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.13·7-s + 2·9-s + 1.37·19-s − 1.96·21-s − 1.92·27-s − 2.22·29-s + 0.538·31-s + 0.986·37-s − 3.20·43-s − 1.75·47-s + 6/7·49-s − 0.824·53-s − 2.38·57-s + 0.781·59-s + 0.384·61-s + 2.26·63-s + 3.29·67-s + 1.42·71-s − 1.05·73-s + 1.01·79-s + 5/3·81-s − 1.97·83-s + 3.85·87-s + 2.54·89-s − 0.933·93-s + 2.13·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.966140408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966140408\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $D_{6}$ | \( 1 + 2 T^{3} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 3 T^{2} - 6 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 8 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 30 T^{2} + 16 T^{3} + 30 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 21 T^{2} - 20 T^{3} + 21 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 45 T^{2} + 8 T^{3} + 45 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 12 T + 120 T^{2} + 702 T^{3} + 120 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 81 T^{2} - 170 T^{3} + 81 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 102 T^{2} - 426 T^{3} + 102 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 66 T^{2} + 156 T^{3} + 66 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 21 T + 255 T^{2} + 2018 T^{3} + 255 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 165 T^{2} + 1104 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 428 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 676 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 138 T^{2} - 199 T^{3} + 138 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 27 T + 423 T^{2} - 4142 T^{3} + 423 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 237 T^{2} - 1664 T^{3} + 237 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 9 T + 186 T^{2} + 1145 T^{3} + 186 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 9 T + 117 T^{2} - 1438 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 237 T^{2} + 1996 T^{3} + 237 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 24 T + 399 T^{2} - 4320 T^{3} + 399 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 21 T + 423 T^{2} - 4310 T^{3} + 423 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.60983891784057640247276350946, −7.05029568154656511348107990406, −6.92473888251530543137341722943, −6.81576699904175026043821434328, −6.42569746679308151542138025645, −6.12942438766381027904746854269, −6.07252424108368229405500390543, −5.61223827824897015082609330792, −5.43439086495657568602450961003, −5.16146882256428853103930810635, −4.92123508138568010411649881347, −4.79411148708188671017900642155, −4.78858960686653143524958829910, −4.14042782948675432263032050991, −3.78040418566812755524785912785, −3.64306213862886545122474549623, −3.49336091705908283054217476976, −2.97330775638594425455364297976, −2.59698633492082307489630334788, −2.06502369109031095256660344301, −1.79016835378878497581560171697, −1.66369132352253404687645980528, −1.16841179027851472316925511544, −0.68722426792356382661648738284, −0.40191075096947912657930693814,
0.40191075096947912657930693814, 0.68722426792356382661648738284, 1.16841179027851472316925511544, 1.66369132352253404687645980528, 1.79016835378878497581560171697, 2.06502369109031095256660344301, 2.59698633492082307489630334788, 2.97330775638594425455364297976, 3.49336091705908283054217476976, 3.64306213862886545122474549623, 3.78040418566812755524785912785, 4.14042782948675432263032050991, 4.78858960686653143524958829910, 4.79411148708188671017900642155, 4.92123508138568010411649881347, 5.16146882256428853103930810635, 5.43439086495657568602450961003, 5.61223827824897015082609330792, 6.07252424108368229405500390543, 6.12942438766381027904746854269, 6.42569746679308151542138025645, 6.81576699904175026043821434328, 6.92473888251530543137341722943, 7.05029568154656511348107990406, 7.60983891784057640247276350946
