| L(s) = 1 | + 2·2-s + 3·5-s − 3·7-s − 2·8-s − 5·9-s + 6·10-s + 8·11-s − 6·14-s + 4·17-s − 10·18-s + 19-s + 16·22-s − 3·23-s − 5·25-s − 2·27-s − 7·29-s + 11·31-s + 8·34-s − 9·35-s + 12·37-s + 2·38-s − 6·40-s + 2·41-s + 13·43-s − 15·45-s − 6·46-s + 19·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.34·5-s − 1.13·7-s − 0.707·8-s − 5/3·9-s + 1.89·10-s + 2.41·11-s − 1.60·14-s + 0.970·17-s − 2.35·18-s + 0.229·19-s + 3.41·22-s − 0.625·23-s − 25-s − 0.384·27-s − 1.29·29-s + 1.97·31-s + 1.37·34-s − 1.52·35-s + 1.97·37-s + 0.324·38-s − 0.948·40-s + 0.312·41-s + 1.98·43-s − 2.23·45-s − 0.884·46-s + 2.77·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.700723510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.700723510\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - p T + p^{2} T^{2} - 3 p T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 3 T + 14 T^{2} - 29 T^{3} + 14 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 8 T + 51 T^{2} - 186 T^{3} + 51 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 43 T^{2} - 6 p T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - T - 2 T^{2} + 155 T^{3} - 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 44 T^{2} + 59 T^{3} + 44 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 66 T^{2} + 411 T^{3} + 66 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 11 T + 112 T^{2} - 617 T^{3} + 112 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 129 T^{2} - 834 T^{3} + 129 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} - 124 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 13 T + 164 T^{2} - 1101 T^{3} + 164 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 19 T + 246 T^{2} - 1923 T^{3} + 246 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - T + 150 T^{2} - 93 T^{3} + 150 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 145 T^{2} - 288 T^{3} + 145 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 211 T^{2} - 1556 T^{3} + 211 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 233 T^{2} + 1592 T^{3} + 233 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 14 T + 159 T^{2} - 1098 T^{3} + 159 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 9 T + 128 T^{2} + 1345 T^{3} + 128 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 13 T + 200 T^{2} - 1869 T^{3} + 200 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - T + 136 T^{2} - 3 T^{3} + 136 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 3 T + 212 T^{2} - 307 T^{3} + 212 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 15 T + 116 T^{2} - 727 T^{3} + 116 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.065867343530271789695503557600, −8.320425187224922935045423226472, −8.295026319821216538636588807231, −7.72575824452561980122975295964, −7.56965636087640590303727796955, −7.30342917669754425945960291208, −6.69735277696923399255664688611, −6.57440497497880168047330685009, −6.08872352672913374333344888314, −6.02494035280313568461482230932, −5.73557710874878805376508265677, −5.63296470084467582923449616624, −5.57247632240827729326083193489, −4.76638939056486511260646642096, −4.58819501182421502292973590935, −4.17126936481034138733956717173, −3.92637381484093191968950008888, −3.70209734211277593721334376515, −3.45785869643102227876086493136, −2.96923469701108296894248486041, −2.42174090614303765318102823003, −2.32912559074355418817301360524, −1.79989071942677762997028862077, −0.854569666052719107087121084131, −0.800333982960221957166513757844,
0.800333982960221957166513757844, 0.854569666052719107087121084131, 1.79989071942677762997028862077, 2.32912559074355418817301360524, 2.42174090614303765318102823003, 2.96923469701108296894248486041, 3.45785869643102227876086493136, 3.70209734211277593721334376515, 3.92637381484093191968950008888, 4.17126936481034138733956717173, 4.58819501182421502292973590935, 4.76638939056486511260646642096, 5.57247632240827729326083193489, 5.63296470084467582923449616624, 5.73557710874878805376508265677, 6.02494035280313568461482230932, 6.08872352672913374333344888314, 6.57440497497880168047330685009, 6.69735277696923399255664688611, 7.30342917669754425945960291208, 7.56965636087640590303727796955, 7.72575824452561980122975295964, 8.295026319821216538636588807231, 8.320425187224922935045423226472, 9.065867343530271789695503557600
