| L(s) = 1 | + 3-s + 4.30·5-s − 7-s + 9-s + 2.56·11-s + 2.81·13-s + 4.30·15-s − 7.96·17-s − 19-s − 21-s + 0.814·23-s + 13.5·25-s + 27-s + 3.48·29-s − 3.38·31-s + 2.56·33-s − 4.30·35-s + 3.75·37-s + 2.81·39-s + 6.60·41-s + 4.30·45-s − 10.2·47-s + 49-s − 7.96·51-s + 0.511·53-s + 11.0·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.92·5-s − 0.377·7-s + 0.333·9-s + 0.773·11-s + 0.780·13-s + 1.11·15-s − 1.93·17-s − 0.229·19-s − 0.218·21-s + 0.169·23-s + 2.70·25-s + 0.192·27-s + 0.647·29-s − 0.607·31-s + 0.446·33-s − 0.727·35-s + 0.616·37-s + 0.450·39-s + 1.03·41-s + 0.641·45-s − 1.49·47-s + 0.142·49-s − 1.11·51-s + 0.0702·53-s + 1.48·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.929681914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.929681914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 4.30T + 5T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 + 7.96T + 17T^{2} \) |
| 23 | \( 1 - 0.814T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 31 | \( 1 + 3.38T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 - 6.60T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 0.511T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 5.02T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 2.01T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 5.66T + 97T^{2} \) |
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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
−8.316108466692559757116499108866, −7.05607063610597319687885680487, −6.43624722763196305801960413550, −6.20975597298241018282218436975, −5.21525221439150932542979468132, −4.41704130327637513812073720345, −3.52722335678043494703445996275, −2.49315212671903643000100927298, −2.02326289895478754166558057429, −1.05796845532890284405195902496,
1.05796845532890284405195902496, 2.02326289895478754166558057429, 2.49315212671903643000100927298, 3.52722335678043494703445996275, 4.41704130327637513812073720345, 5.21525221439150932542979468132, 6.20975597298241018282218436975, 6.43624722763196305801960413550, 7.05607063610597319687885680487, 8.316108466692559757116499108866
