| L(s) = 1 | + 1.76·2-s − 4.86·4-s − 13.0·5-s − 22.7·8-s − 23.0·10-s − 52.9·11-s − 71.7·13-s − 1.32·16-s − 106.·17-s + 53.9·19-s + 63.5·20-s − 93.6·22-s − 18.6·23-s + 45.3·25-s − 126.·26-s − 261.·29-s − 122.·31-s + 179.·32-s − 188.·34-s + 278.·37-s + 95.4·38-s + 297.·40-s + 31.3·41-s − 347.·43-s + 257.·44-s − 33.0·46-s − 542.·47-s + ⋯ |
| L(s) = 1 | + 0.625·2-s − 0.608·4-s − 1.16·5-s − 1.00·8-s − 0.730·10-s − 1.45·11-s − 1.52·13-s − 0.0207·16-s − 1.52·17-s + 0.651·19-s + 0.710·20-s − 0.907·22-s − 0.169·23-s + 0.362·25-s − 0.956·26-s − 1.67·29-s − 0.708·31-s + 0.993·32-s − 0.951·34-s + 1.23·37-s + 0.407·38-s + 1.17·40-s + 0.119·41-s − 1.23·43-s + 0.883·44-s − 0.105·46-s − 1.68·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01317466474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01317466474\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.76T + 8T^{2} \) |
| 5 | \( 1 + 13.0T + 125T^{2} \) |
| 11 | \( 1 + 52.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 53.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 18.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 261.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 278.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 31.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 347.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 542.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 257.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 315.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 138.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 397.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 843.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 872.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 554.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 297.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 102.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 515.T + 9.12e5T^{2} \) |
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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
−9.317213490035143149494365981821, −8.259501248012529337970693464824, −7.70565706394104275366536602385, −6.91309891299673494411277694810, −5.57818149490698119003788171828, −4.91384506573205949595512456958, −4.23946213575936815079065119819, −3.28124396318783369939309500597, −2.31512589412499989767793667148, −0.04865050536869086106741872234,
0.04865050536869086106741872234, 2.31512589412499989767793667148, 3.28124396318783369939309500597, 4.23946213575936815079065119819, 4.91384506573205949595512456958, 5.57818149490698119003788171828, 6.91309891299673494411277694810, 7.70565706394104275366536602385, 8.259501248012529337970693464824, 9.317213490035143149494365981821
