L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 2·13-s − 8·17-s − 6·19-s − 21-s + 8·23-s − 5·25-s − 27-s − 6·29-s + 10·31-s + 33-s + 2·37-s − 2·39-s − 8·41-s − 4·43-s + 2·47-s + 49-s + 8·51-s + 2·53-s + 6·57-s − 8·59-s + 2·61-s + 63-s + 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 1.94·17-s − 1.37·19-s − 0.218·21-s + 1.66·23-s − 25-s − 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s − 1.24·41-s − 0.609·43-s + 0.291·47-s + 1/7·49-s + 1.12·51-s + 0.274·53-s + 0.794·57-s − 1.04·59-s + 0.256·61-s + 0.125·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.721318483216356043846903841700, −8.243314571231184041063756885284, −7.04906935731541709957659667985, −6.52612451347175351673765101894, −5.64991830703907805418699018236, −4.67739315792929506623467661343, −4.10559754758205212771034018100, −2.69477535422873391316325182968, −1.60554009680005977870138036167, 0,
1.60554009680005977870138036167, 2.69477535422873391316325182968, 4.10559754758205212771034018100, 4.67739315792929506623467661343, 5.64991830703907805418699018236, 6.52612451347175351673765101894, 7.04906935731541709957659667985, 8.243314571231184041063756885284, 8.721318483216356043846903841700