| L(s) = 1 | + 4.12·2-s + 9·4-s − 3.05·5-s + 6.68·7-s + 4.12·8-s − 12.5·10-s + 32.2·11-s + 27.5·14-s − 55·16-s + 28.8·17-s + 101.·19-s − 27.4·20-s + 132.·22-s + 118.·23-s − 115.·25-s + 60.1·28-s − 160.·29-s + 38.0·31-s − 259.·32-s + 118.·34-s − 20.3·35-s + 327.·37-s + 417.·38-s − 12.5·40-s + 56.0·41-s + 127.·43-s + 290.·44-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 1.12·4-s − 0.272·5-s + 0.360·7-s + 0.182·8-s − 0.397·10-s + 0.883·11-s + 0.526·14-s − 0.859·16-s + 0.411·17-s + 1.22·19-s − 0.306·20-s + 1.28·22-s + 1.07·23-s − 0.925·25-s + 0.405·28-s − 1.02·29-s + 0.220·31-s − 1.43·32-s + 0.600·34-s − 0.0984·35-s + 1.45·37-s + 1.78·38-s − 0.0497·40-s + 0.213·41-s + 0.453·43-s + 0.994·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.316941404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.316941404\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.12T + 8T^{2} \) |
| 5 | \( 1 + 3.05T + 125T^{2} \) |
| 7 | \( 1 - 6.68T + 343T^{2} \) |
| 11 | \( 1 - 32.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 28.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 38.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 56.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 517.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 695.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 656.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 57.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 389.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 901.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 687.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.75e3T + 9.12e5T^{2} \) |
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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
−9.212832663145339013765578606444, −8.117453886663768047521028597185, −7.26294560069896469971942129078, −6.47430993480727375950192274851, −5.56607778051098132518112166994, −4.95908746764577473682108019122, −3.95645860451255534333727268761, −3.41901348728940541595164949784, −2.27338577481491736693551178231, −0.934167160143925575694819025422,
0.934167160143925575694819025422, 2.27338577481491736693551178231, 3.41901348728940541595164949784, 3.95645860451255534333727268761, 4.95908746764577473682108019122, 5.56607778051098132518112166994, 6.47430993480727375950192274851, 7.26294560069896469971942129078, 8.117453886663768047521028597185, 9.212832663145339013765578606444
