| L(s) = 1 | + 81·3-s + 2.46e3·7-s + 6.56e3·9-s − 9.44e3·11-s − 4.16e4·13-s − 1.96e5·17-s + 2.89e5·19-s + 1.99e5·21-s + 1.26e6·23-s + 5.31e5·27-s − 4.90e6·29-s − 7.73e6·31-s − 7.64e5·33-s + 1.85e6·37-s − 3.37e6·39-s − 4.73e6·41-s + 1.69e7·43-s + 3.48e7·47-s − 3.42e7·49-s − 1.59e7·51-s − 1.04e7·53-s + 2.34e7·57-s − 4.47e7·59-s − 9.91e7·61-s + 1.61e7·63-s + 1.10e8·67-s + 1.02e8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.387·7-s + 1/3·9-s − 0.194·11-s − 0.404·13-s − 0.571·17-s + 0.509·19-s + 0.223·21-s + 0.944·23-s + 0.192·27-s − 1.28·29-s − 1.50·31-s − 0.112·33-s + 0.162·37-s − 0.233·39-s − 0.261·41-s + 0.756·43-s + 1.04·47-s − 0.849·49-s − 0.330·51-s − 0.182·53-s + 0.294·57-s − 0.481·59-s − 0.916·61-s + 0.129·63-s + 0.670·67-s + 0.545·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{4} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 352 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 9444 T + p^{9} T^{2} \) |
| 13 | \( 1 + 41630 T + p^{9} T^{2} \) |
| 17 | \( 1 + 196866 T + p^{9} T^{2} \) |
| 19 | \( 1 - 289700 T + p^{9} T^{2} \) |
| 23 | \( 1 - 1267104 T + p^{9} T^{2} \) |
| 29 | \( 1 + 4906146 T + p^{9} T^{2} \) |
| 31 | \( 1 + 7730176 T + p^{9} T^{2} \) |
| 37 | \( 1 - 50146 p T + p^{9} T^{2} \) |
| 41 | \( 1 + 4736310 T + p^{9} T^{2} \) |
| 43 | \( 1 - 16965244 T + p^{9} T^{2} \) |
| 47 | \( 1 - 34814616 T + p^{9} T^{2} \) |
| 53 | \( 1 + 10476822 T + p^{9} T^{2} \) |
| 59 | \( 1 + 44778276 T + p^{9} T^{2} \) |
| 61 | \( 1 + 99115618 T + p^{9} T^{2} \) |
| 67 | \( 1 - 110662516 T + p^{9} T^{2} \) |
| 71 | \( 1 + 51816840 T + p^{9} T^{2} \) |
| 73 | \( 1 + 165642698 T + p^{9} T^{2} \) |
| 79 | \( 1 + 36181312 T + p^{9} T^{2} \) |
| 83 | \( 1 + 594531276 T + p^{9} T^{2} \) |
| 89 | \( 1 - 299211450 T + p^{9} T^{2} \) |
| 97 | \( 1 + 302501762 T + p^{9} T^{2} \) |
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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
−9.494788277165612500259709662145, −8.873383513851307691700616595945, −7.72423591773811503650324022208, −7.08100475052842357986025764864, −5.68095867737527362047463478191, −4.67339416434866492846778710253, −3.54469004345477206123620259406, −2.44084186173089774014596195565, −1.41203225500095578298002908974, 0,
1.41203225500095578298002908974, 2.44084186173089774014596195565, 3.54469004345477206123620259406, 4.67339416434866492846778710253, 5.68095867737527362047463478191, 7.08100475052842357986025764864, 7.72423591773811503650324022208, 8.873383513851307691700616595945, 9.494788277165612500259709662145
