| L(s) = 1 | − 1.38·2-s + 1.10·3-s − 0.0894·4-s − 3.03·5-s − 1.52·6-s − 3.07·7-s + 2.88·8-s − 1.77·9-s + 4.19·10-s − 0.193·11-s − 0.0988·12-s + 2.08·13-s + 4.24·14-s − 3.35·15-s − 3.81·16-s + 17-s + 2.45·18-s − 3.76·19-s + 0.271·20-s − 3.39·21-s + 0.268·22-s − 3.35·23-s + 3.19·24-s + 4.22·25-s − 2.88·26-s − 5.28·27-s + 0.274·28-s + ⋯ |
| L(s) = 1 | − 0.977·2-s + 0.637·3-s − 0.0447·4-s − 1.35·5-s − 0.623·6-s − 1.16·7-s + 1.02·8-s − 0.593·9-s + 1.32·10-s − 0.0584·11-s − 0.0285·12-s + 0.579·13-s + 1.13·14-s − 0.866·15-s − 0.953·16-s + 0.242·17-s + 0.579·18-s − 0.864·19-s + 0.0607·20-s − 0.740·21-s + 0.0571·22-s − 0.699·23-s + 0.651·24-s + 0.845·25-s − 0.566·26-s − 1.01·27-s + 0.0519·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4823478900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4823478900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 3 | \( 1 - 1.10T + 3T^{2} \) |
| 5 | \( 1 + 3.03T + 5T^{2} \) |
| 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 + 0.193T + 11T^{2} \) |
| 13 | \( 1 - 2.08T + 13T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 37 | \( 1 - 3.91T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 - 1.19T + 47T^{2} \) |
| 53 | \( 1 - 7.96T + 53T^{2} \) |
| 61 | \( 1 - 6.10T + 61T^{2} \) |
| 67 | \( 1 + 4.46T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 4.22T + 73T^{2} \) |
| 79 | \( 1 + 7.77T + 79T^{2} \) |
| 83 | \( 1 - 0.248T + 83T^{2} \) |
| 89 | \( 1 - 1.18T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−9.722025625819742683267199540931, −9.053084289942989602720502426159, −8.259660784130286384383465267512, −7.920285540464155593178487743333, −6.92337945408559707285038659875, −5.88074270750603026317025083126, −4.29652311389798140128953150347, −3.68548649178296646635287908211, −2.55767324482640920064566370267, −0.58486674700265822320542789462,
0.58486674700265822320542789462, 2.55767324482640920064566370267, 3.68548649178296646635287908211, 4.29652311389798140128953150347, 5.88074270750603026317025083126, 6.92337945408559707285038659875, 7.920285540464155593178487743333, 8.259660784130286384383465267512, 9.053084289942989602720502426159, 9.722025625819742683267199540931
