| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 2·7-s − 8-s + 9-s + 2·10-s + 11-s + 12-s − 2·14-s − 2·15-s + 16-s + 17-s − 18-s − 2·19-s − 2·20-s + 2·21-s − 22-s + 2·23-s − 24-s − 25-s + 27-s + 2·28-s + 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.447·20-s + 0.436·21-s − 0.213·22-s + 0.417·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.917061036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917061036\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.05162559669486, −12.49511933250294, −12.12411404019283, −11.58475129336510, −11.23967425553959, −10.83314989920573, −10.12527344179117, −9.873274147432277, −9.235936906189586, −8.618894439271303, −8.394969992422150, −7.963843038992092, −7.554678959164396, −6.930206125278582, −6.607956580078312, −5.931469947385489, −5.086402057337385, −4.803336436532301, −4.045376007088847, −3.597415066602565, −3.094122635817112, −2.312330158024433, −1.826883905007866, −1.140006403990610, −0.4607082954156123,
0.4607082954156123, 1.140006403990610, 1.826883905007866, 2.312330158024433, 3.094122635817112, 3.597415066602565, 4.045376007088847, 4.803336436532301, 5.086402057337385, 5.931469947385489, 6.607956580078312, 6.930206125278582, 7.554678959164396, 7.963843038992092, 8.394969992422150, 8.618894439271303, 9.235936906189586, 9.873274147432277, 10.12527344179117, 10.83314989920573, 11.23967425553959, 11.58475129336510, 12.12411404019283, 12.49511933250294, 13.05162559669486
