| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 13-s + 16-s − 6·17-s − 4·19-s − 20-s − 8·23-s + 25-s − 26-s + 6·29-s − 8·31-s + 32-s − 6·34-s − 10·37-s − 4·38-s − 40-s − 6·41-s − 4·43-s − 8·46-s − 7·49-s + 50-s − 52-s + 10·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.277·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.196·26-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 1.64·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 1.17·46-s − 49-s + 0.141·50-s − 0.138·52-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.82893158306808, −13.37599099334165, −13.01809568639976, −12.30515578165856, −12.10786383502636, −11.61474000806278, −11.08508247510918, −10.47809856102880, −10.32295879965222, −9.601956168415217, −8.834055118372495, −8.564233661801426, −8.065976397635295, −7.355866178916456, −6.933236234534184, −6.480243207523348, −5.985821601293042, −5.316909266814312, −4.809116186916890, −4.224278118673797, −3.924798480886260, −3.233432929631779, −2.586756614012863, −1.937637548775127, −1.518176281688234, 0, 0,
1.518176281688234, 1.937637548775127, 2.586756614012863, 3.233432929631779, 3.924798480886260, 4.224278118673797, 4.809116186916890, 5.316909266814312, 5.985821601293042, 6.480243207523348, 6.933236234534184, 7.355866178916456, 8.065976397635295, 8.564233661801426, 8.834055118372495, 9.601956168415217, 10.32295879965222, 10.47809856102880, 11.08508247510918, 11.61474000806278, 12.10786383502636, 12.30515578165856, 13.01809568639976, 13.37599099334165, 13.82893158306808
