L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 2·13-s + 15-s − 2·17-s + 4·19-s − 23-s + 25-s − 27-s − 2·29-s − 8·31-s + 4·33-s + 6·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s − 8·47-s + 2·51-s − 10·53-s + 4·55-s − 4·57-s + 12·59-s − 14·61-s − 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 0.280·51-s − 1.37·53-s + 0.539·55-s − 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.45940478826838, −13.28140822265028, −12.60906281073700, −12.37038075907874, −11.67610664933800, −11.13487597929351, −10.97195106608322, −10.50580028407954, −9.770376040286798, −9.407023494986354, −8.898691996585409, −8.088818383610595, −7.776400349773578, −7.476889801237290, −6.621750225552850, −6.328794173555263, −5.623798017348767, −5.117510549213858, −4.819060861861065, −3.980770290000432, −3.555383241285883, −2.904852281262416, −2.204903986524376, −1.518914263968274, −0.6819592745268475, 0,
0.6819592745268475, 1.518914263968274, 2.204903986524376, 2.904852281262416, 3.555383241285883, 3.980770290000432, 4.819060861861065, 5.117510549213858, 5.623798017348767, 6.328794173555263, 6.621750225552850, 7.476889801237290, 7.776400349773578, 8.088818383610595, 8.898691996585409, 9.407023494986354, 9.770376040286798, 10.50580028407954, 10.97195106608322, 11.13487597929351, 11.67610664933800, 12.37038075907874, 12.60906281073700, 13.28140822265028, 13.45940478826838