| L(s) = 1 | − 3.36·2-s + 3.31·4-s − 2.29·5-s + 15.7·8-s + 7.73·10-s − 44.4·11-s − 20.7·13-s − 79.5·16-s − 13.2·17-s − 62.0·19-s − 7.61·20-s + 149.·22-s + 90.1·23-s − 119.·25-s + 69.8·26-s − 100.·29-s − 94.5·31-s + 141.·32-s + 44.6·34-s + 196.·37-s + 208.·38-s − 36.2·40-s + 253.·41-s − 111.·43-s − 147.·44-s − 303.·46-s + 67.7·47-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 0.414·4-s − 0.205·5-s + 0.696·8-s + 0.244·10-s − 1.21·11-s − 0.442·13-s − 1.24·16-s − 0.189·17-s − 0.749·19-s − 0.0851·20-s + 1.44·22-s + 0.817·23-s − 0.957·25-s + 0.526·26-s − 0.645·29-s − 0.547·31-s + 0.781·32-s + 0.225·34-s + 0.873·37-s + 0.890·38-s − 0.143·40-s + 0.964·41-s − 0.393·43-s − 0.504·44-s − 0.972·46-s + 0.210·47-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
| \(L(2)\) |
\(\approx\) |
\(0.4009123139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4009123139\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.36T + 8T^{2} \) |
| 5 | \( 1 + 2.29T + 125T^{2} \) |
| 11 | \( 1 + 44.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 62.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 90.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 358.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 92.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 432.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 695.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 20.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 871.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 233.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 168.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.14e3T + 9.12e5T^{2} \) |
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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
−9.279872115648345644102172585744, −8.475649437521100963303264185676, −7.70979845335107404213045358016, −7.28354953906816882142880311915, −6.06939929573574391029796091567, −5.01672629678486810696361036025, −4.17471287823295620746720732657, −2.77160027493984575583104685336, −1.76112432882888312823824299991, −0.37375584359369891671941513857,
0.37375584359369891671941513857, 1.76112432882888312823824299991, 2.77160027493984575583104685336, 4.17471287823295620746720732657, 5.01672629678486810696361036025, 6.06939929573574391029796091567, 7.28354953906816882142880311915, 7.70979845335107404213045358016, 8.475649437521100963303264185676, 9.279872115648345644102172585744
