| L(s) = 1 | − 2-s − 4-s − 5-s + 3·7-s + 8-s + 10-s + 7·13-s − 3·14-s − 16-s − 17-s + 8·19-s + 20-s + 2·23-s − 7·25-s − 7·26-s − 3·28-s − 7·29-s − 2·31-s + 32-s + 34-s − 3·35-s − 3·37-s − 8·38-s − 40-s − 13·41-s + 8·43-s − 2·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s + 0.316·10-s + 1.94·13-s − 0.801·14-s − 1/4·16-s − 0.242·17-s + 1.83·19-s + 0.223·20-s + 0.417·23-s − 7/5·25-s − 1.37·26-s − 0.566·28-s − 1.29·29-s − 0.359·31-s + 0.176·32-s + 0.171·34-s − 0.507·35-s − 0.493·37-s − 1.29·38-s − 0.158·40-s − 2.03·41-s + 1.21·43-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.379004807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.379004807\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + T + 8 T^{2} + 11 T^{3} + 8 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 7 T + 51 T^{2} - 184 T^{3} + 51 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + T + 44 T^{2} + 35 T^{3} + 44 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 63 T^{2} - 260 T^{3} + 63 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T - 9 T^{2} + 60 T^{3} - 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 99 T^{2} + 408 T^{3} + 99 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 132 T^{3} + 65 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 107 T^{2} + 214 T^{3} + 107 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 13 T + 163 T^{2} + 1098 T^{3} + 163 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 58 T^{2} - 266 T^{3} + 58 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 110 T^{2} - 530 T^{3} + 110 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 21 T + 299 T^{2} - 2522 T^{3} + 299 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 622 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 12 T + 165 T^{2} - 1172 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 50 T^{2} - 852 T^{3} + 50 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 179 T^{2} + 76 T^{3} + 179 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T - 7 T^{2} - 1044 T^{3} - 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 18 T + 209 T^{2} - 1636 T^{3} + 209 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 78 T^{2} + 64 T^{3} + 78 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - T + 188 T^{2} - 251 T^{3} + 188 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 25 T + 315 T^{2} + 2968 T^{3} + 315 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.02098577536126392120722147338, −6.79745884846357387142791527071, −6.56934948232797859229671098707, −6.15395162640469822584308006228, −6.14455579827531752801754494083, −5.63155403376443342310899142277, −5.51460919517325450567253663895, −5.26496533129620963729649557169, −5.17114512883715914405550758343, −5.03085676595265624505111716775, −4.40188471031630479817861266736, −4.17753259805900457180779837411, −4.13620062316898863243895961219, −3.73723917988996103679826159001, −3.61633066854771695604746068431, −3.44571361077046265682679334400, −2.90901324575992591043964907439, −2.83156736140648020759787145366, −2.08888940217803112141084603907, −2.06396636976327026727194185873, −1.75194926047249391797269885618, −1.41179674072325599413952426371, −0.965183611406877851471093011788, −0.56001567491050870885591686983, −0.54435252349942297671831475387,
0.54435252349942297671831475387, 0.56001567491050870885591686983, 0.965183611406877851471093011788, 1.41179674072325599413952426371, 1.75194926047249391797269885618, 2.06396636976327026727194185873, 2.08888940217803112141084603907, 2.83156736140648020759787145366, 2.90901324575992591043964907439, 3.44571361077046265682679334400, 3.61633066854771695604746068431, 3.73723917988996103679826159001, 4.13620062316898863243895961219, 4.17753259805900457180779837411, 4.40188471031630479817861266736, 5.03085676595265624505111716775, 5.17114512883715914405550758343, 5.26496533129620963729649557169, 5.51460919517325450567253663895, 5.63155403376443342310899142277, 6.14455579827531752801754494083, 6.15395162640469822584308006228, 6.56934948232797859229671098707, 6.79745884846357387142791527071, 7.02098577536126392120722147338
