| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s − 3·5-s + 9·6-s + 3·7-s + 10·8-s + 6·9-s − 9·10-s + 11-s + 18·12-s + 9·14-s − 9·15-s + 15·16-s + 12·17-s + 18·18-s + 4·19-s − 18·20-s + 9·21-s + 3·22-s + 16·23-s + 30·24-s − 2·25-s + 10·27-s + 18·28-s + 13·29-s − 27·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s + 1.13·7-s + 3.53·8-s + 2·9-s − 2.84·10-s + 0.301·11-s + 5.19·12-s + 2.40·14-s − 2.32·15-s + 15/4·16-s + 2.91·17-s + 4.24·18-s + 0.917·19-s − 4.02·20-s + 1.96·21-s + 0.639·22-s + 3.33·23-s + 6.12·24-s − 2/5·25-s + 1.92·27-s + 3.40·28-s + 2.41·29-s − 4.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(27.70873680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.70873680\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 3 T + 11 T^{2} + 31 T^{3} + 11 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 17 T^{2} - 43 T^{3} + 17 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - T + 17 T^{2} - 35 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 12 T + 71 T^{2} - 304 T^{3} + 71 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 4 T + 25 T^{2} - 88 T^{3} + 25 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 16 T + 145 T^{2} - 840 T^{3} + 145 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 13 T + 99 T^{2} - 531 T^{3} + 99 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 9 T + 71 T^{2} + 529 T^{3} + 71 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 12 T + 131 T^{2} + 896 T^{3} + 131 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 14 T + 179 T^{2} + 1204 T^{3} + 179 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 8 T + 85 T^{2} + 8 p T^{3} + 85 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 4 T + 109 T^{2} - 312 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 15 T + 87 T^{2} - 343 T^{3} + 87 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 9 T + 197 T^{2} + 1075 T^{3} + 197 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 10 T + 207 T^{2} + 1228 T^{3} + 207 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 308 T^{3} + 17 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 6 T + 141 T^{2} - 748 T^{3} + 141 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 5 T + 197 T^{2} - 717 T^{3} + 197 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 5 T + 33 T^{2} - 679 T^{3} + 33 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 25 T^{2} - 315 T^{3} + 25 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 10 T + 291 T^{2} + 1788 T^{3} + 291 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 7 T + 277 T^{2} - 1351 T^{3} + 277 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.845218238706019495384329167901, −8.359873835161890311639533985353, −8.175257530052979246896455844341, −8.005866956444706997993098420887, −7.58572510315508435458140344308, −7.41243678578284885324547662895, −7.32414158715904383617297145133, −6.86070391608869888729693546101, −6.62800985141558394163668824549, −6.48446213074067317772895634041, −5.49796056869188660020166585707, −5.46765643659823242065605418549, −5.34167414345820092380154285060, −4.83993880642181214309290634499, −4.74552887541911428073683140399, −4.36982047888807898836709379631, −3.78826683053152034517761369152, −3.60348136083605367309432240033, −3.39677022724100594602251071794, −3.14822827482958216267996048640, −2.90383236945367784484430532902, −2.50625964745840427597499722266, −1.61111813697039943435414343039, −1.44507304698032299892028334665, −1.18236672390134377279201613725,
1.18236672390134377279201613725, 1.44507304698032299892028334665, 1.61111813697039943435414343039, 2.50625964745840427597499722266, 2.90383236945367784484430532902, 3.14822827482958216267996048640, 3.39677022724100594602251071794, 3.60348136083605367309432240033, 3.78826683053152034517761369152, 4.36982047888807898836709379631, 4.74552887541911428073683140399, 4.83993880642181214309290634499, 5.34167414345820092380154285060, 5.46765643659823242065605418549, 5.49796056869188660020166585707, 6.48446213074067317772895634041, 6.62800985141558394163668824549, 6.86070391608869888729693546101, 7.32414158715904383617297145133, 7.41243678578284885324547662895, 7.58572510315508435458140344308, 8.005866956444706997993098420887, 8.175257530052979246896455844341, 8.359873835161890311639533985353, 8.845218238706019495384329167901
