| L(s) = 1 | + 1.02e3·4-s + 1.09e4·7-s − 1.41e5·13-s + 1.04e6·16-s + 4.92e6·19-s + 9.76e6·25-s + 1.11e7·28-s + 4.93e7·31-s − 4.08e7·37-s − 2.82e8·43-s − 1.63e8·49-s − 1.45e8·52-s − 1.35e9·61-s + 1.07e9·64-s + 2.69e9·67-s + 1.95e9·73-s + 5.04e9·76-s − 6.05e9·79-s − 1.54e9·91-s − 1.52e10·97-s + 1.00e10·100-s − 1.82e10·103-s − 1.76e10·109-s + 1.14e10·112-s + ⋯ |
| L(s) = 1 | + 4-s + 0.648·7-s − 0.382·13-s + 16-s + 1.98·19-s + 25-s + 0.648·28-s + 1.72·31-s − 0.589·37-s − 1.92·43-s − 0.578·49-s − 0.382·52-s − 1.60·61-s + 64-s + 1.99·67-s + 0.944·73-s + 1.98·76-s − 1.96·79-s − 0.248·91-s − 1.78·97-s + 100-s − 1.57·103-s − 1.14·109-s + 0.648·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.606564216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.606564216\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 5 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 7 | \( 1 - 10907 T + p^{10} T^{2} \) |
| 11 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 13 | \( 1 + 141961 T + p^{10} T^{2} \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( 1 - 4926251 T + p^{10} T^{2} \) |
| 23 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 29 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 31 | \( 1 - 49326674 T + p^{10} T^{2} \) |
| 37 | \( 1 + 40895593 T + p^{10} T^{2} \) |
| 41 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 43 | \( 1 + 282780982 T + p^{10} T^{2} \) |
| 47 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 53 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 59 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 61 | \( 1 + 1354266001 T + p^{10} T^{2} \) |
| 67 | \( 1 - 2698325411 T + p^{10} T^{2} \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( 1 - 1957684943 T + p^{10} T^{2} \) |
| 79 | \( 1 + 6059886949 T + p^{10} T^{2} \) |
| 83 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 89 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 97 | \( 1 + 15296411593 T + p^{10} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.10897233771988936681046773664, −13.91320064904530394078692702111, −12.18445339564894320163557943259, −11.29411118258827257415831146295, −9.938722119307850696693413414500, −8.074648144393432000214785922438, −6.83341164141748903694757616143, −5.15467648076097719150934558906, −2.97693944742217042948623518665, −1.31679181013728795472017017929,
1.31679181013728795472017017929, 2.97693944742217042948623518665, 5.15467648076097719150934558906, 6.83341164141748903694757616143, 8.074648144393432000214785922438, 9.938722119307850696693413414500, 11.29411118258827257415831146295, 12.18445339564894320163557943259, 13.91320064904530394078692702111, 15.10897233771988936681046773664
