| L(s) = 1 | + 9.10·5-s + 1.25i·7-s + (−6.90 + 8.55i)11-s + 4.36i·13-s − 25.9i·17-s + 22.6i·19-s + 1.15·23-s + 57.9·25-s + 37.4i·29-s + 14.5·31-s + 11.4i·35-s − 10.5·37-s + 43.0i·41-s + 56.7i·43-s + 56.2·47-s + ⋯ |
| L(s) = 1 | + 1.82·5-s + 0.179i·7-s + (−0.628 + 0.778i)11-s + 0.335i·13-s − 1.52i·17-s + 1.19i·19-s + 0.0501·23-s + 2.31·25-s + 1.29i·29-s + 0.467·31-s + 0.327i·35-s − 0.283·37-s + 1.05i·41-s + 1.31i·43-s + 1.19·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.778i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.741040687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.741040687\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (6.90 - 8.55i)T \) |
| good | 5 | \( 1 - 9.10T + 25T^{2} \) |
| 7 | \( 1 - 1.25iT - 49T^{2} \) |
| 13 | \( 1 - 4.36iT - 169T^{2} \) |
| 17 | \( 1 + 25.9iT - 289T^{2} \) |
| 19 | \( 1 - 22.6iT - 361T^{2} \) |
| 23 | \( 1 - 1.15T + 529T^{2} \) |
| 29 | \( 1 - 37.4iT - 841T^{2} \) |
| 31 | \( 1 - 14.5T + 961T^{2} \) |
| 37 | \( 1 + 10.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 43.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 56.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 56.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.62T + 2.80e3T^{2} \) |
| 59 | \( 1 - 70.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 39.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 107.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 41.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 72.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 52.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 75.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 64.4T + 9.40e3T^{2} \) |
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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
−9.402666928926107350519598048832, −8.840900444365344411969959691631, −7.64351510917830668569982294115, −6.85342304000668587097364861182, −6.01718631653473557417726739885, −5.29038334518020049578328237263, −4.62116012180885968205131244557, −3.01113782239127945026710240541, −2.23474042234835799911208312953, −1.28374974724869598579413787354,
0.75762638715453902253695227938, 2.04568870568879903486398718355, 2.76658444042400652197379710127, 4.06210653559120880830302797074, 5.30715361299467511921847511026, 5.79597968929128041437630686006, 6.48483230757266826142717726106, 7.46935336519727563598077828685, 8.629274698373941181822079944760, 8.990651213638358746029020815024
