| L(s) = 1 | − 1.96e4·3-s + 2.62e5·4-s + 7.75e7·7-s + 3.87e8·9-s − 5.15e9·12-s − 7.19e9·13-s + 6.87e10·16-s + 3.08e11·19-s − 1.52e12·21-s + 3.81e12·25-s − 7.62e12·27-s + 2.03e13·28-s − 5.00e13·31-s + 1.01e14·36-s − 2.32e13·37-s + 1.41e14·39-s − 7.30e14·43-s − 1.35e15·48-s + 4.38e15·49-s − 1.88e15·52-s − 6.07e15·57-s − 9.48e15·61-s + 3.00e16·63-s + 1.80e16·64-s − 4.17e16·67-s − 2.99e16·73-s − 7.50e16·75-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 1.92·7-s + 9-s − 12-s − 0.678·13-s + 16-s + 0.956·19-s − 1.92·21-s + 25-s − 27-s + 1.92·28-s − 1.89·31-s + 36-s − 0.178·37-s + 0.678·39-s − 1.45·43-s − 48-s + 2.69·49-s − 0.678·52-s − 0.956·57-s − 0.811·61-s + 1.92·63-s + 64-s − 1.53·67-s − 0.508·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(1.709371284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709371284\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{9} T \) |
| good | 2 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 5 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 7 | \( 1 - 77549186 T + p^{18} T^{2} \) |
| 11 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 13 | \( 1 + 7197541846 T + p^{18} T^{2} \) |
| 17 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 19 | \( 1 - 308559680858 T + p^{18} T^{2} \) |
| 23 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 29 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 31 | \( 1 + 50018992173358 T + p^{18} T^{2} \) |
| 37 | \( 1 + 23240947030054 T + p^{18} T^{2} \) |
| 41 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 43 | \( 1 + 730385642547286 T + p^{18} T^{2} \) |
| 47 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 53 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 59 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 61 | \( 1 + 9487161099916918 T + p^{18} T^{2} \) |
| 67 | \( 1 + 41747295001607494 T + p^{18} T^{2} \) |
| 71 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 73 | \( 1 + 29908998244279726 T + p^{18} T^{2} \) |
| 79 | \( 1 - 140655567501204338 T + p^{18} T^{2} \) |
| 83 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 89 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 97 | \( 1 - 140873967896062466 T + p^{18} T^{2} \) |
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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
−21.77854477956418141031559305259, −20.48082782010022655435302256267, −18.09964511267097949481694223101, −16.69456671997730106254388482187, −14.87218570319768166537777733204, −11.93961335683160982463994356361, −10.84222534393047283285836018935, −7.40714737867680867729692971268, −5.18282147434873618398849598591, −1.54937755958486096374629988010,
1.54937755958486096374629988010, 5.18282147434873618398849598591, 7.40714737867680867729692971268, 10.84222534393047283285836018935, 11.93961335683160982463994356361, 14.87218570319768166537777733204, 16.69456671997730106254388482187, 18.09964511267097949481694223101, 20.48082782010022655435302256267, 21.77854477956418141031559305259
