| L(s) = 1 | + 3·2-s − 2·3-s + 6·4-s − 5-s − 6·6-s − 2·7-s + 10·8-s − 3·10-s − 12·12-s − 13-s − 6·14-s + 2·15-s + 15·16-s − 5·17-s − 3·19-s − 6·20-s + 4·21-s + 2·23-s − 20·24-s − 8·25-s − 3·26-s + 4·27-s − 12·28-s + 29-s + 6·30-s + 21·32-s − 15·34-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.15·3-s + 3·4-s − 0.447·5-s − 2.44·6-s − 0.755·7-s + 3.53·8-s − 0.948·10-s − 3.46·12-s − 0.277·13-s − 1.60·14-s + 0.516·15-s + 15/4·16-s − 1.21·17-s − 0.688·19-s − 1.34·20-s + 0.872·21-s + 0.417·23-s − 4.08·24-s − 8/5·25-s − 0.588·26-s + 0.769·27-s − 2.26·28-s + 0.185·29-s + 1.09·30-s + 3.71·32-s − 2.57·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 4 T^{2} + 4 T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + T + 9 T^{2} + 12 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 2 p T^{2} + 24 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 31 T^{2} + 22 T^{3} + 31 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 5 T + 3 p T^{2} + 162 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 62 T^{2} - 88 T^{3} + 62 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - T + 68 T^{2} - 83 T^{3} + 68 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 53 T^{2} - 64 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 23 T + 268 T^{2} + 1973 T^{3} + 268 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 101 T^{2} + 496 T^{3} + 101 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 101 T^{2} - 312 T^{3} + 101 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 62 T^{2} + 56 T^{3} + 62 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 5 T + 100 T^{2} - 567 T^{3} + 100 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 18 T + 4 p T^{2} + 2168 T^{3} + 4 p^{2} T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 344 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 89 T^{2} + 816 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 203 T^{2} - 8 T^{3} + 203 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 155 T^{2} - 948 T^{3} + 155 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 223 T^{2} + 640 T^{3} + 223 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 235 T^{2} - 672 T^{3} + 235 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 7 T + 125 T^{2} - 1388 T^{3} + 125 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 7 T + 283 T^{2} - 1342 T^{3} + 283 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.58780688345613118699659718382, −7.33510801555942068517436000301, −6.94018233601734903851122908215, −6.74512114132157739061666052072, −6.65335813309611636240154420846, −6.45491054849334739547649896690, −6.30417001790904337294603493733, −5.98566139788073694277556462028, −5.62825749145019296587310160223, −5.45297460513828538521207937878, −5.20156982604335854050014148043, −5.09580002707600189041794908821, −4.75539056320370041955401718305, −4.43605384307644530211188196081, −4.41598122049440959941774048961, −3.85067331679107192728838315724, −3.55328145921969728760777201990, −3.48030523334629787585003376169, −3.44894263179504292792870571708, −2.72233440437581715290583620067, −2.49701711623085969963799812444, −2.42662095296569579447857421198, −1.77115620948359312062960589331, −1.53529571695812710730601693159, −1.28726585956882012974288788040, 0, 0, 0,
1.28726585956882012974288788040, 1.53529571695812710730601693159, 1.77115620948359312062960589331, 2.42662095296569579447857421198, 2.49701711623085969963799812444, 2.72233440437581715290583620067, 3.44894263179504292792870571708, 3.48030523334629787585003376169, 3.55328145921969728760777201990, 3.85067331679107192728838315724, 4.41598122049440959941774048961, 4.43605384307644530211188196081, 4.75539056320370041955401718305, 5.09580002707600189041794908821, 5.20156982604335854050014148043, 5.45297460513828538521207937878, 5.62825749145019296587310160223, 5.98566139788073694277556462028, 6.30417001790904337294603493733, 6.45491054849334739547649896690, 6.65335813309611636240154420846, 6.74512114132157739061666052072, 6.94018233601734903851122908215, 7.33510801555942068517436000301, 7.58780688345613118699659718382
