| L(s) = 1 | − 2·2-s + 2·4-s + 4·7-s − 8-s + 8·11-s + 4·13-s − 8·14-s − 4·16-s − 4·17-s − 2·19-s − 16·22-s − 6·23-s + 25-s − 8·26-s + 8·28-s − 3·29-s + 6·31-s + 8·32-s + 8·34-s + 8·37-s + 4·38-s + 2·41-s − 4·43-s + 16·44-s + 12·46-s + 12·47-s − 4·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.51·7-s − 0.353·8-s + 2.41·11-s + 1.10·13-s − 2.13·14-s − 16-s − 0.970·17-s − 0.458·19-s − 3.41·22-s − 1.25·23-s + 1/5·25-s − 1.56·26-s + 1.51·28-s − 0.557·29-s + 1.07·31-s + 1.41·32-s + 1.37·34-s + 1.31·37-s + 0.648·38-s + 0.312·41-s − 0.609·43-s + 2.41·44-s + 1.76·46-s + 1.75·47-s − 4/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17779581 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17779581 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.081512276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.081512276\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T + p T^{2} + T^{3} + p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T^{2} - 8 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 20 T^{2} - 48 T^{3} + 20 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 8 T + 48 T^{2} - 180 T^{3} + 48 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 32 T^{2} - 6 p T^{3} + 32 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 24 T^{2} + 42 T^{3} + 24 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 37 T^{2} + 92 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 65 T^{2} + 244 T^{3} + 65 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 340 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 584 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 23 T^{2} - 220 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 88 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 132 T^{2} - 912 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 55 T^{2} + 600 T^{3} + 55 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 20 T + 285 T^{2} - 2472 T^{3} + 285 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 167 T^{2} - 432 T^{3} + 167 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 144 T^{2} + 52 T^{3} + 144 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 14 T + 153 T^{2} - 1572 T^{3} + 153 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 1160 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 177 T^{2} + 92 T^{3} + 177 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 221 T^{2} - 1120 T^{3} + 221 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 136 T^{2} - 1350 T^{3} + 136 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 4 T + 219 T^{2} - 880 T^{3} + 219 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.74080013280750863418459716338, −10.50317658005145421373983048345, −9.875108252286056924441838603473, −9.735640367801854909548350458050, −9.309956059886718691277450343830, −9.051956494889258457298395819590, −8.827804660270956228612202293003, −8.464601111909517501056914192685, −8.110151513408090971118370390688, −8.108279550078177918900370083806, −7.47710922814531512634163957415, −7.11114218892230118223003304842, −6.64955494139607739085014912079, −6.37470607736323218614690014798, −6.26459468618223473350599710979, −5.69070465123847630004924937603, −5.07477201768043971528596570792, −4.54613745325072059167389947720, −4.33020598851083300111558312439, −3.85551656049954225721533006205, −3.58988054192390223323751359248, −2.43102823933071948428892909782, −2.09940579716822489259011887500, −1.49442799016086090491750613038, −0.943727449094005939339096958814,
0.943727449094005939339096958814, 1.49442799016086090491750613038, 2.09940579716822489259011887500, 2.43102823933071948428892909782, 3.58988054192390223323751359248, 3.85551656049954225721533006205, 4.33020598851083300111558312439, 4.54613745325072059167389947720, 5.07477201768043971528596570792, 5.69070465123847630004924937603, 6.26459468618223473350599710979, 6.37470607736323218614690014798, 6.64955494139607739085014912079, 7.11114218892230118223003304842, 7.47710922814531512634163957415, 8.108279550078177918900370083806, 8.110151513408090971118370390688, 8.464601111909517501056914192685, 8.827804660270956228612202293003, 9.051956494889258457298395819590, 9.309956059886718691277450343830, 9.735640367801854909548350458050, 9.875108252286056924441838603473, 10.50317658005145421373983048345, 10.74080013280750863418459716338
