| L(s) = 1 | + 4-s − 3·5-s − 7-s + 2·8-s + 11-s + 3·16-s + 17-s − 6·19-s − 3·20-s + 7·23-s + 6·25-s − 28-s − 18·29-s − 6·31-s + 4·32-s + 3·35-s − 13·37-s − 6·40-s + 41-s + 44-s − 18·47-s − 4·49-s − 11·53-s − 3·55-s − 2·56-s − 8·59-s + 9·61-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.34·5-s − 0.377·7-s + 0.707·8-s + 0.301·11-s + 3/4·16-s + 0.242·17-s − 1.37·19-s − 0.670·20-s + 1.45·23-s + 6/5·25-s − 0.188·28-s − 3.34·29-s − 1.07·31-s + 0.707·32-s + 0.507·35-s − 2.13·37-s − 0.948·40-s + 0.156·41-s + 0.150·44-s − 2.62·47-s − 4/7·49-s − 1.51·53-s − 0.404·55-s − 0.267·56-s − 1.04·59-s + 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.410744066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410744066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{6}$ | \( 1 - T^{2} - p T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 17 T^{2} - 38 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 19 T^{2} + 42 T^{3} + 19 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 164 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 53 T^{2} - 194 T^{3} + 53 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 340 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 13 T + 3 p T^{2} + 646 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 91 T^{2} - 6 T^{3} + 91 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 128 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 18 T + 221 T^{2} + 1756 T^{3} + 221 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 11 T + 167 T^{2} + 1162 T^{3} + 167 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 129 T^{2} + 816 T^{3} + 129 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 9 T + 71 T^{2} - 254 T^{3} + 71 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 137 T^{2} + 408 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 11 T + 237 T^{2} + 1530 T^{3} + 237 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 119 T^{2} - 532 T^{3} + 119 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 5 T + 189 T^{2} - 854 T^{3} + 189 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 201 T^{2} - 1200 T^{3} + 201 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 11 T + 275 T^{2} - 1954 T^{3} + 275 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 25 T + 467 T^{2} - 5094 T^{3} + 467 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.12251816563488248674210462797, −6.68742209196189899712721193103, −6.56788517453826253386185867952, −6.28171295583257598830389680491, −6.18005278189003148039676546224, −5.86320476513349069527742152007, −5.46026962233504032840138052066, −5.08830992475884281237378550272, −5.07085030556174251929790487843, −5.05116780834779799763758690595, −4.42649979898793636951498585216, −4.30683406640865254030282928184, −4.08051272858261244008613495911, −3.61417193376530627179986912671, −3.55738271259470257153557531356, −3.30236901786848724988133736313, −3.08859590613385750249500443272, −3.00657416303661360575931611227, −2.13303332097309088791309906640, −2.07772235932025980745268000985, −1.83378156851657311104279269259, −1.49862903044605799206655307661, −1.22374048099086340152974269000, −0.47680356865281767206152257177, −0.28343701912383238950952337329,
0.28343701912383238950952337329, 0.47680356865281767206152257177, 1.22374048099086340152974269000, 1.49862903044605799206655307661, 1.83378156851657311104279269259, 2.07772235932025980745268000985, 2.13303332097309088791309906640, 3.00657416303661360575931611227, 3.08859590613385750249500443272, 3.30236901786848724988133736313, 3.55738271259470257153557531356, 3.61417193376530627179986912671, 4.08051272858261244008613495911, 4.30683406640865254030282928184, 4.42649979898793636951498585216, 5.05116780834779799763758690595, 5.07085030556174251929790487843, 5.08830992475884281237378550272, 5.46026962233504032840138052066, 5.86320476513349069527742152007, 6.18005278189003148039676546224, 6.28171295583257598830389680491, 6.56788517453826253386185867952, 6.68742209196189899712721193103, 7.12251816563488248674210462797
