| L(s) = 1 | + 5.45·2-s + 21.7·4-s + 19.3·5-s + 74.6·8-s + 105.·10-s − 11.2·11-s + 46.3·13-s + 233.·16-s − 97.6·17-s + 98.9·19-s + 419.·20-s − 61.1·22-s − 138.·23-s + 249.·25-s + 252.·26-s − 180.·29-s + 31.9·31-s + 674.·32-s − 532.·34-s − 205.·37-s + 539.·38-s + 1.44e3·40-s + 234.·41-s − 320.·43-s − 243.·44-s − 753.·46-s − 312.·47-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.71·4-s + 1.73·5-s + 3.30·8-s + 3.33·10-s − 0.307·11-s + 0.988·13-s + 3.64·16-s − 1.39·17-s + 1.19·19-s + 4.69·20-s − 0.592·22-s − 1.25·23-s + 1.99·25-s + 1.90·26-s − 1.15·29-s + 0.184·31-s + 3.72·32-s − 2.68·34-s − 0.912·37-s + 2.30·38-s + 5.71·40-s + 0.892·41-s − 1.13·43-s − 0.833·44-s − 2.41·46-s − 0.970·47-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
| \(L(2)\) |
\(\approx\) |
\(11.32978267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.32978267\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.45T + 8T^{2} \) |
| 5 | \( 1 - 19.3T + 125T^{2} \) |
| 11 | \( 1 + 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 180.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 31.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 205.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 320.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 312.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 53.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 400.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 97.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 257.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 253.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 889.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 647.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 673.T + 9.12e5T^{2} \) |
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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
−9.517290857551499338066026279580, −8.315812974165553967563384872361, −7.05688596511353251145038072990, −6.42032231571924739413219226076, −5.70829340412164963848690854490, −5.24680080187716122156228637759, −4.19594796719648209110241181773, −3.19690931310824252268215342043, −2.20592866073550392957286851640, −1.56723881923926150421203915090,
1.56723881923926150421203915090, 2.20592866073550392957286851640, 3.19690931310824252268215342043, 4.19594796719648209110241181773, 5.24680080187716122156228637759, 5.70829340412164963848690854490, 6.42032231571924739413219226076, 7.05688596511353251145038072990, 8.315812974165553967563384872361, 9.517290857551499338066026279580
