| L(s) = 1 | + 0.438·2-s − 7.80·4-s + 17.8·5-s + 5.43·7-s − 6.93·8-s + 7.80·10-s + 22.4·11-s + 2.38·14-s + 59.4·16-s − 67.9·17-s − 80.8·19-s − 139.·20-s + 9.83·22-s − 140.·23-s + 192.·25-s − 42.4·28-s + 106.·29-s − 276.·31-s + 81.5·32-s − 29.8·34-s + 96.8·35-s − 4.29·37-s − 35.4·38-s − 123.·40-s − 227.·41-s + 27.5·43-s − 175.·44-s + ⋯ |
| L(s) = 1 | + 0.155·2-s − 0.975·4-s + 1.59·5-s + 0.293·7-s − 0.306·8-s + 0.246·10-s + 0.614·11-s + 0.0455·14-s + 0.928·16-s − 0.969·17-s − 0.975·19-s − 1.55·20-s + 0.0952·22-s − 1.27·23-s + 1.53·25-s − 0.286·28-s + 0.683·29-s − 1.59·31-s + 0.450·32-s − 0.150·34-s + 0.467·35-s − 0.0190·37-s − 0.151·38-s − 0.487·40-s − 0.867·41-s + 0.0976·43-s − 0.599·44-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.438T + 8T^{2} \) |
| 5 | \( 1 - 17.8T + 125T^{2} \) |
| 7 | \( 1 - 5.43T + 343T^{2} \) |
| 11 | \( 1 - 22.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 67.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 276.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 4.29T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 27.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 67.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 291.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 663.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 425.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 152.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 117.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 202.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 336.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 718.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 759.T + 9.12e5T^{2} \) |
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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
−8.799722706409645660572593637243, −8.219434246748475562559429762658, −6.81244566606679427048644429924, −6.14569818452916036340595705413, −5.38702614432198955271145704166, −4.58137958742649290985586532051, −3.69199647175063466005914692404, −2.28497017755463231702982176320, −1.49815554504421215419380241640, 0,
1.49815554504421215419380241640, 2.28497017755463231702982176320, 3.69199647175063466005914692404, 4.58137958742649290985586532051, 5.38702614432198955271145704166, 6.14569818452916036340595705413, 6.81244566606679427048644429924, 8.219434246748475562559429762658, 8.799722706409645660572593637243
