| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s − 3·11-s − 12-s + 13-s − 2·14-s − 15-s + 16-s + 18-s − 3·19-s + 20-s + 2·21-s − 3·22-s − 9·23-s − 24-s + 25-s + 26-s − 27-s − 2·28-s + 29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.688·19-s + 0.223·20-s + 0.436·21-s − 0.639·22-s − 1.87·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.68337644697343, −12.32289158301201, −12.01780396078097, −11.36198697646503, −10.90535389021346, −10.50160949387770, −10.08944802795502, −9.807021528468636, −9.115728344106250, −8.656745332248042, −7.937614370355382, −7.712593296459263, −6.976465993968077, −6.595708894010051, −6.046833079373270, −5.834440185115053, −5.371780033668158, −4.715013620022718, −4.304503037666052, −3.755748435501653, −3.213389177197128, −2.616046056700224, −2.074541834561346, −1.605257473374102, −0.6457106125555989, 0,
0.6457106125555989, 1.605257473374102, 2.074541834561346, 2.616046056700224, 3.213389177197128, 3.755748435501653, 4.304503037666052, 4.715013620022718, 5.371780033668158, 5.834440185115053, 6.046833079373270, 6.595708894010051, 6.976465993968077, 7.712593296459263, 7.937614370355382, 8.656745332248042, 9.115728344106250, 9.807021528468636, 10.08944802795502, 10.50160949387770, 10.90535389021346, 11.36198697646503, 12.01780396078097, 12.32289158301201, 12.68337644697343
