L(s) = 1 | − 2.30·2-s − 1.30·3-s + 3.30·4-s + 2.30·5-s + 3·6-s − 0.302·7-s − 3.00·8-s − 1.30·9-s − 5.30·10-s + 3·11-s − 4.30·12-s − 3.30·13-s + 0.697·14-s − 3·15-s + 0.302·16-s + 3.00·18-s + 5.90·19-s + 7.60·20-s + 0.394·21-s − 6.90·22-s + 2.30·23-s + 3.90·24-s + 0.302·25-s + 7.60·26-s + 5.60·27-s − 1.00·28-s + 9.90·29-s + ⋯ |
L(s) = 1 | − 1.62·2-s − 0.752·3-s + 1.65·4-s + 1.02·5-s + 1.22·6-s − 0.114·7-s − 1.06·8-s − 0.434·9-s − 1.67·10-s + 0.904·11-s − 1.24·12-s − 0.916·13-s + 0.186·14-s − 0.774·15-s + 0.0756·16-s + 0.707·18-s + 1.35·19-s + 1.70·20-s + 0.0860·21-s − 1.47·22-s + 0.480·23-s + 0.797·24-s + 0.0605·25-s + 1.49·26-s + 1.07·27-s − 0.188·28-s + 1.83·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5414261164\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5414261164\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 + 0.302T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 - 2.30T + 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 - 0.605T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 2.09T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 8.81T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 3.21T + 71T^{2} \) |
| 73 | \( 1 - 0.394T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−11.58355337208950469837100617743, −10.59076176250504581571078577670, −9.766922515757910540581931739630, −9.255094647672446488567327467373, −8.153567162983770907321287060676, −6.93807261932316016050049257441, −6.19311897312072138745257282442, −5.00510947505132922630092617013, −2.63998034217618756816259058982, −1.04969752797638415086499215154,
1.04969752797638415086499215154, 2.63998034217618756816259058982, 5.00510947505132922630092617013, 6.19311897312072138745257282442, 6.93807261932316016050049257441, 8.153567162983770907321287060676, 9.255094647672446488567327467373, 9.766922515757910540581931739630, 10.59076176250504581571078577670, 11.58355337208950469837100617743