L(s) = 1 | + 2-s − 3.43·3-s + 4-s − 2.28·5-s − 3.43·6-s − 3.85·7-s + 8-s + 8.81·9-s − 2.28·10-s + 0.361·11-s − 3.43·12-s − 4.38·13-s − 3.85·14-s + 7.84·15-s + 16-s + 1.54·17-s + 8.81·18-s − 7.18·19-s − 2.28·20-s + 13.2·21-s + 0.361·22-s + 6.47·23-s − 3.43·24-s + 0.211·25-s − 4.38·26-s − 19.9·27-s − 3.85·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.98·3-s + 0.5·4-s − 1.02·5-s − 1.40·6-s − 1.45·7-s + 0.353·8-s + 2.93·9-s − 0.721·10-s + 0.108·11-s − 0.992·12-s − 1.21·13-s − 1.03·14-s + 2.02·15-s + 0.250·16-s + 0.374·17-s + 2.07·18-s − 1.64·19-s − 0.510·20-s + 2.89·21-s + 0.0769·22-s + 1.35·23-s − 0.701·24-s + 0.0422·25-s − 0.860·26-s − 3.84·27-s − 0.729·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 4001 | \( 1+O(T) \) |
good | 3 | \( 1 + 3.43T + 3T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 - 0.361T + 11T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 - 1.54T + 17T^{2} \) |
| 19 | \( 1 + 7.18T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 - 9.01T + 31T^{2} \) |
| 37 | \( 1 + 0.704T + 37T^{2} \) |
| 41 | \( 1 + 1.54T + 41T^{2} \) |
| 43 | \( 1 - 6.07T + 43T^{2} \) |
| 47 | \( 1 - 6.06T + 47T^{2} \) |
| 53 | \( 1 - 2.49T + 53T^{2} \) |
| 59 | \( 1 - 9.64T + 59T^{2} \) |
| 61 | \( 1 + 5.89T + 61T^{2} \) |
| 67 | \( 1 - 9.47T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 1.80T + 97T^{2} \) |
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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
−7.09988742403313891948701821339, −6.71324916011434198107316590883, −6.06188725038045797756557141338, −5.46544445762560253585200546206, −4.60077090181906498834046025340, −4.22291491553475578199726539748, −3.41331182809365153328172767286, −2.31910107267963884458754788028, −0.809637002682085496100135516037, 0,
0.809637002682085496100135516037, 2.31910107267963884458754788028, 3.41331182809365153328172767286, 4.22291491553475578199726539748, 4.60077090181906498834046025340, 5.46544445762560253585200546206, 6.06188725038045797756557141338, 6.71324916011434198107316590883, 7.09988742403313891948701821339