| L(s) = 1 | − 1.02e3·2-s + 6.24e4·3-s + 1.04e6·4-s + 9.76e6·5-s − 6.38e7·6-s − 5.48e8·7-s − 1.07e9·8-s + 4.07e8·9-s − 1.00e10·10-s + 6.54e10·12-s + 5.61e11·14-s + 6.09e11·15-s + 1.09e12·16-s − 4.16e11·18-s + 1.02e13·20-s − 3.42e13·21-s + 8.14e13·23-s − 6.70e13·24-s + 9.53e13·25-s − 1.92e14·27-s − 5.74e14·28-s + 6.16e14·29-s − 6.24e14·30-s − 1.12e15·32-s − 5.35e15·35-s + 4.27e14·36-s − 1.04e16·40-s + ⋯ |
| L(s) = 1 | − 2-s + 1.05·3-s + 4-s + 5-s − 1.05·6-s − 1.94·7-s − 8-s + 0.116·9-s − 10-s + 1.05·12-s + 1.94·14-s + 1.05·15-s + 16-s − 0.116·18-s + 20-s − 2.05·21-s + 1.96·23-s − 1.05·24-s + 25-s − 0.933·27-s − 1.94·28-s + 1.46·29-s − 1.05·30-s − 32-s − 1.94·35-s + 0.116·36-s − 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(1.819537105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.819537105\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{10} T \) |
| 5 | \( 1 - p^{10} T \) |
| good | 3 | \( 1 - 62402 T + p^{20} T^{2} \) |
| 7 | \( 1 + 548205998 T + p^{20} T^{2} \) |
| 11 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 13 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 17 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 19 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 23 | \( 1 - 81485142477202 T + p^{20} T^{2} \) |
| 29 | \( 1 - 616275178208402 T + p^{20} T^{2} \) |
| 31 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 37 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 41 | \( 1 - 17687539525192802 T + p^{20} T^{2} \) |
| 43 | \( 1 + 6802335769227998 T + p^{20} T^{2} \) |
| 47 | \( 1 - 95811025658740402 T + p^{20} T^{2} \) |
| 53 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 59 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 61 | \( 1 + 341772141346522798 T + p^{20} T^{2} \) |
| 67 | \( 1 + 1845116665718910398 T + p^{20} T^{2} \) |
| 71 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 73 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 79 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 83 | \( 1 - 1338748174791257602 T + p^{20} T^{2} \) |
| 89 | \( 1 - 61250728984437679202 T + p^{20} T^{2} \) |
| 97 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
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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
−13.68085064017681530557369466393, −12.59812882505746042961407338934, −10.49977600621063195602378925246, −9.428739867320512906113238306145, −8.871801417724440977752323859941, −7.08459141790127367277398125964, −6.01363393519654802435781522460, −3.13599638992141414543441958616, −2.52798389045355229091900856991, −0.820105177051512981058988643118,
0.820105177051512981058988643118, 2.52798389045355229091900856991, 3.13599638992141414543441958616, 6.01363393519654802435781522460, 7.08459141790127367277398125964, 8.871801417724440977752323859941, 9.428739867320512906113238306145, 10.49977600621063195602378925246, 12.59812882505746042961407338934, 13.68085064017681530557369466393
