| L(s) = 1 | + 2-s − 1.06·3-s + 4-s + 3.41·5-s − 1.06·6-s − 0.896·7-s + 8-s − 1.86·9-s + 3.41·10-s + 2.68·11-s − 1.06·12-s + 0.0919·13-s − 0.896·14-s − 3.63·15-s + 16-s + 0.524·17-s − 1.86·18-s + 19-s + 3.41·20-s + 0.956·21-s + 2.68·22-s + 3.82·23-s − 1.06·24-s + 6.63·25-s + 0.0919·26-s + 5.18·27-s − 0.896·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.615·3-s + 0.5·4-s + 1.52·5-s − 0.435·6-s − 0.339·7-s + 0.353·8-s − 0.620·9-s + 1.07·10-s + 0.809·11-s − 0.307·12-s + 0.0254·13-s − 0.239·14-s − 0.939·15-s + 0.250·16-s + 0.127·17-s − 0.438·18-s + 0.229·19-s + 0.762·20-s + 0.208·21-s + 0.572·22-s + 0.796·23-s − 0.217·24-s + 1.32·25-s + 0.0180·26-s + 0.998·27-s − 0.169·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.666235982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.666235982\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.06T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 + 0.896T + 7T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 - 0.0919T + 13T^{2} \) |
| 17 | \( 1 - 0.524T + 17T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + 7.05T + 31T^{2} \) |
| 37 | \( 1 - 6.20T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 - 5.86T + 47T^{2} \) |
| 53 | \( 1 + 7.97T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 - 6.14T + 61T^{2} \) |
| 67 | \( 1 + 6.95T + 67T^{2} \) |
| 71 | \( 1 - 9.45T + 71T^{2} \) |
| 73 | \( 1 - 7.50T + 73T^{2} \) |
| 79 | \( 1 - 6.17T + 79T^{2} \) |
| 83 | \( 1 - 1.77T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 + 0.805T + 97T^{2} \) |
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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
−7.63545490394742222859991837342, −6.58682877585165745633476310871, −6.44703763516887964176362525269, −5.71148120070830205606846307945, −5.23154578723422908120196761764, −4.51631796665921680214823433034, −3.41601992989966261291626333218, −2.74522719158000398006648582780, −1.85456547016237780173640446581, −0.905814540507301698688935500074,
0.905814540507301698688935500074, 1.85456547016237780173640446581, 2.74522719158000398006648582780, 3.41601992989966261291626333218, 4.51631796665921680214823433034, 5.23154578723422908120196761764, 5.71148120070830205606846307945, 6.44703763516887964176362525269, 6.58682877585165745633476310871, 7.63545490394742222859991837342
