| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 4·11-s + 12-s + 2·13-s + 15-s + 16-s + 18-s − 19-s + 20-s − 4·22-s − 4·23-s + 24-s + 25-s + 2·26-s + 27-s − 6·29-s + 30-s − 4·31-s + 32-s − 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.852·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s − 0.718·31-s + 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.711610455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.711610455\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.36981442054086, −12.87210679665281, −12.60204435596100, −11.84888106806058, −11.35138217928223, −10.95453097759279, −10.35357702667586, −9.980358145006087, −9.521949779714386, −8.913148388727350, −8.197544489616301, −8.084880994325083, −7.403043061984757, −6.908335012227295, −6.270027656974338, −5.869468687174325, −5.301082995207037, −4.810546597390753, −4.295735912646020, −3.458129571749598, −3.296752884030072, −2.558957268326425, −1.821320554613053, −1.692391030049330, −0.4291032283857462,
0.4291032283857462, 1.692391030049330, 1.821320554613053, 2.558957268326425, 3.296752884030072, 3.458129571749598, 4.295735912646020, 4.810546597390753, 5.301082995207037, 5.869468687174325, 6.270027656974338, 6.908335012227295, 7.403043061984757, 8.084880994325083, 8.197544489616301, 8.913148388727350, 9.521949779714386, 9.980358145006087, 10.35357702667586, 10.95453097759279, 11.35138217928223, 11.84888106806058, 12.60204435596100, 12.87210679665281, 13.36981442054086
