| L(s) = 1 | − 64·4-s + 286·7-s − 506·13-s − 2.11e4·19-s − 1.56e4·25-s − 1.83e4·28-s − 3.52e4·31-s − 1.78e5·37-s − 1.11e5·43-s + 1.17e5·49-s + 3.23e4·52-s + 4.20e5·61-s + 2.62e5·64-s − 1.72e5·67-s + 1.27e6·73-s + 1.35e6·76-s + 2.04e5·79-s − 1.44e5·91-s + 5.64e4·97-s + 1.00e6·100-s − 1.12e6·103-s − 4.34e6·109-s − 1.77e6·121-s + 2.25e6·124-s + 127-s + 131-s − 6.05e6·133-s + ⋯ |
| L(s) = 1 | − 4-s + 0.833·7-s − 0.230·13-s − 3.08·19-s − 25-s − 0.833·28-s − 1.18·31-s − 3.52·37-s − 1.40·43-s + 49-s + 0.230·52-s + 1.85·61-s + 64-s − 0.574·67-s + 3.28·73-s + 3.08·76-s + 0.415·79-s − 0.192·91-s + 0.0618·97-s + 100-s − 1.03·103-s − 3.35·109-s − 121-s + 1.18·124-s − 2.57·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.2176608814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2176608814\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )( 1 + p^{3} T + p^{6} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )( 1 + p^{3} T + p^{6} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 683 T + p^{6} T^{2} )( 1 + 397 T + p^{6} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )( 1 + p^{3} T + p^{6} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3527 T + p^{6} T^{2} )( 1 + 4033 T + p^{6} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 10582 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )( 1 + p^{3} T + p^{6} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )( 1 + p^{3} T + p^{6} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 23939 T + p^{6} T^{2} )( 1 + 59221 T + p^{6} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 89206 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )( 1 + p^{3} T + p^{6} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 42587 T + p^{6} T^{2} )( 1 + 153973 T + p^{6} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )( 1 + p^{3} T + p^{6} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )( 1 + p^{3} T + p^{6} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 357839 T + p^{6} T^{2} )( 1 - 62999 T + p^{6} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 412523 T + p^{6} T^{2} )( 1 + 585397 T + p^{6} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 638066 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 937691 T + p^{6} T^{2} )( 1 + 733069 T + p^{6} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )( 1 + p^{3} T + p^{6} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1608263 T + p^{6} T^{2} )( 1 + 1551817 T + p^{6} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.64761775743518499023999947156, −12.80837273592003777307981380067, −12.51594357608557916505826408948, −11.83955507305472634257739474092, −11.20015848646792680251567178276, −10.43197203104563639867193335447, −10.40875893246579428867332986672, −9.303114437296269617519942390450, −8.917897484162901537898876398709, −8.300452241556915263950583023571, −8.038010923848080228951327438028, −6.85682960341544215736144500300, −6.57019542454504169642404306460, −5.21271796889171959105212281543, −5.19834768057936481162369018214, −4.00195459327551411917969903251, −3.87556032731495333838559028486, −2.23526932620029805615265735536, −1.72810013514798661504621029329, −0.16879197340998936023880176114,
0.16879197340998936023880176114, 1.72810013514798661504621029329, 2.23526932620029805615265735536, 3.87556032731495333838559028486, 4.00195459327551411917969903251, 5.19834768057936481162369018214, 5.21271796889171959105212281543, 6.57019542454504169642404306460, 6.85682960341544215736144500300, 8.038010923848080228951327438028, 8.300452241556915263950583023571, 8.917897484162901537898876398709, 9.303114437296269617519942390450, 10.40875893246579428867332986672, 10.43197203104563639867193335447, 11.20015848646792680251567178276, 11.83955507305472634257739474092, 12.51594357608557916505826408948, 12.80837273592003777307981380067, 13.64761775743518499023999947156
