| L(s) = 1 | − 3·2-s + 6·4-s + 5·7-s − 10·8-s − 5·9-s − 3·11-s − 15·14-s + 15·16-s + 4·17-s + 15·18-s + 13·19-s + 9·22-s + 2·27-s + 30·28-s − 14·29-s − 6·31-s − 21·32-s − 12·34-s − 30·36-s + 5·37-s − 39·38-s − 2·41-s + 6·43-s − 18·44-s + 7·47-s + 49-s + 9·53-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 1.88·7-s − 3.53·8-s − 5/3·9-s − 0.904·11-s − 4.00·14-s + 15/4·16-s + 0.970·17-s + 3.53·18-s + 2.98·19-s + 1.91·22-s + 0.384·27-s + 5.66·28-s − 2.59·29-s − 1.07·31-s − 3.71·32-s − 2.05·34-s − 5·36-s + 0.821·37-s − 6.32·38-s − 0.312·41-s + 0.914·43-s − 2.71·44-s + 1.02·47-s + 1/7·49-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.697602195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697602195\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5 T + 24 T^{2} - 65 T^{3} + 24 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} + 35 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 134 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 13 T + 100 T^{2} - 499 T^{3} + 100 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 29 T^{2} + 76 T^{3} + 29 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 14 T + 115 T^{2} + 660 T^{3} + 115 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 35 T^{2} + 154 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 5 T + 82 T^{2} - 263 T^{3} + 82 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 79 T^{2} + 144 T^{3} + 79 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 - 7 T + 152 T^{2} - 659 T^{3} + 152 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 152 T^{2} - 941 T^{3} + 152 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 145 T^{2} + 672 T^{3} + 145 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 75 T^{2} + 2 p T^{3} + 75 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 233 T^{2} - 1624 T^{3} + 233 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 20 T + 253 T^{2} - 2376 T^{3} + 253 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 93 T^{2} - 1146 T^{3} + 93 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 195 T^{2} + 26 p T^{3} + 195 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 121 T^{2} - 618 T^{3} + 121 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 7 T + 274 T^{2} + 1227 T^{3} + 274 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 26 T + 493 T^{2} - 5510 T^{3} + 493 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.11857661384513822633912849641, −6.77064472304152220740816050498, −6.53568778001151075341615522120, −6.34784589497958688186672494187, −5.71075163220453676624978367068, −5.66425509036892587932663429677, −5.53975427309502727244197572570, −5.35605105176659356675291823823, −5.21670787465782614003935748129, −5.13928329718459054665094260757, −4.56321768244380160702980194755, −4.07004804331544984966465980337, −4.05756252718051871435919484196, −3.45628063324341093371679032317, −3.29529743443471423943595909338, −3.27242434011752150917844370854, −2.57606842189634129244226578282, −2.51940274073992422630755727015, −2.49936113475998148990189835117, −1.80515593654174152997646255339, −1.55912956108049909702411675917, −1.53393777951295792797140442393, −0.949272838457246876165405949611, −0.61738396197743771127397873162, −0.39621005525475193402328147338,
0.39621005525475193402328147338, 0.61738396197743771127397873162, 0.949272838457246876165405949611, 1.53393777951295792797140442393, 1.55912956108049909702411675917, 1.80515593654174152997646255339, 2.49936113475998148990189835117, 2.51940274073992422630755727015, 2.57606842189634129244226578282, 3.27242434011752150917844370854, 3.29529743443471423943595909338, 3.45628063324341093371679032317, 4.05756252718051871435919484196, 4.07004804331544984966465980337, 4.56321768244380160702980194755, 5.13928329718459054665094260757, 5.21670787465782614003935748129, 5.35605105176659356675291823823, 5.53975427309502727244197572570, 5.66425509036892587932663429677, 5.71075163220453676624978367068, 6.34784589497958688186672494187, 6.53568778001151075341615522120, 6.77064472304152220740816050498, 7.11857661384513822633912849641
