| L(s) = 1 | + 2·3-s − 9-s + 8·11-s − 8·13-s − 2·17-s + 3·19-s − 4·23-s − 4·27-s − 10·29-s − 4·31-s + 16·33-s − 20·37-s − 16·39-s − 2·41-s − 4·43-s − 5·49-s − 4·51-s − 16·53-s + 6·57-s + 20·59-s − 2·61-s + 2·67-s − 8·69-s + 4·71-s − 2·73-s − 2·81-s − 32·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/3·9-s + 2.41·11-s − 2.21·13-s − 0.485·17-s + 0.688·19-s − 0.834·23-s − 0.769·27-s − 1.85·29-s − 0.718·31-s + 2.78·33-s − 3.28·37-s − 2.56·39-s − 0.312·41-s − 0.609·43-s − 5/7·49-s − 0.560·51-s − 2.19·53-s + 0.794·57-s + 2.60·59-s − 0.256·61-s + 0.244·67-s − 0.963·69-s + 0.474·71-s − 0.234·73-s − 2/9·81-s − 3.51·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 160 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 51 T^{2} + 212 T^{3} + 51 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} - 36 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 61 T^{2} + 168 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 45 T^{2} + 184 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 20 T + 235 T^{2} + 1724 T^{3} + 235 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 87 T^{2} + 60 T^{3} + 87 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T - 15 T^{2} - 248 T^{3} - 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 125 T^{2} + 16 T^{3} + 125 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 235 T^{2} + 1788 T^{3} + 235 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 20 T + 289 T^{2} - 2520 T^{3} + 289 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 99 T^{2} + 476 T^{3} + 99 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 125 T^{2} - 384 T^{3} + 125 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 133 T^{2} - 504 T^{3} + 133 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2 T + 199 T^{2} + 284 T^{3} + 199 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 45 T^{2} + 160 T^{3} + 45 p T^{4} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 32 T + 577 T^{2} + 6384 T^{3} + 577 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 135 T^{2} + 324 T^{3} + 135 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 20 T + 231 T^{2} + 2132 T^{3} + 231 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.20587034560771748946361471141, −7.11573431876900778730197185024, −6.90902746982280937632358761314, −6.76374503752299927766229964076, −6.43956051456864462532405657304, −6.25286391815367029348990602654, −5.95063211006358843994494040469, −5.49912509100216548242609182848, −5.32471567192947358463819477022, −5.32041547949961066432201412079, −4.89844708967095191010746352855, −4.85229315040019103784762671500, −4.21145781029959199165844173990, −4.00318498725935834869856463843, −3.90672671260561047441199816075, −3.80946680596220026338718833048, −3.29329174240933181629753615829, −3.11307684828708222691684918830, −2.97922040990560593098819031330, −2.45138281893738901978881517751, −2.31612856328929913919517548575, −2.00348846800412125682859589359, −1.58929000581405292299627888140, −1.44298431569799403367677144005, −1.20741229090836664048535132932, 0, 0, 0,
1.20741229090836664048535132932, 1.44298431569799403367677144005, 1.58929000581405292299627888140, 2.00348846800412125682859589359, 2.31612856328929913919517548575, 2.45138281893738901978881517751, 2.97922040990560593098819031330, 3.11307684828708222691684918830, 3.29329174240933181629753615829, 3.80946680596220026338718833048, 3.90672671260561047441199816075, 4.00318498725935834869856463843, 4.21145781029959199165844173990, 4.85229315040019103784762671500, 4.89844708967095191010746352855, 5.32041547949961066432201412079, 5.32471567192947358463819477022, 5.49912509100216548242609182848, 5.95063211006358843994494040469, 6.25286391815367029348990602654, 6.43956051456864462532405657304, 6.76374503752299927766229964076, 6.90902746982280937632358761314, 7.11573431876900778730197185024, 7.20587034560771748946361471141
