L(s) = 1 | − 2.74·2-s + 0.238·3-s + 5.55·4-s − 0.412·5-s − 0.655·6-s − 2.19·7-s − 9.76·8-s − 2.94·9-s + 1.13·10-s − 2.64·11-s + 1.32·12-s + 3.93·13-s + 6.02·14-s − 0.0983·15-s + 15.7·16-s − 5.33·17-s + 8.08·18-s − 1.03·19-s − 2.29·20-s − 0.522·21-s + 7.25·22-s + 1.82·23-s − 2.32·24-s − 4.82·25-s − 10.8·26-s − 1.41·27-s − 12.1·28-s + ⋯ |
L(s) = 1 | − 1.94·2-s + 0.137·3-s + 2.77·4-s − 0.184·5-s − 0.267·6-s − 0.828·7-s − 3.45·8-s − 0.981·9-s + 0.358·10-s − 0.796·11-s + 0.382·12-s + 1.09·13-s + 1.60·14-s − 0.0253·15-s + 3.93·16-s − 1.29·17-s + 1.90·18-s − 0.237·19-s − 0.512·20-s − 0.114·21-s + 1.54·22-s + 0.381·23-s − 0.475·24-s − 0.965·25-s − 2.12·26-s − 0.272·27-s − 2.30·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2120987215\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2120987215\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 - 0.238T + 3T^{2} \) |
| 5 | \( 1 + 0.412T + 5T^{2} \) |
| 7 | \( 1 + 2.19T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 + 5.33T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 - 0.842T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 - 8.32T + 37T^{2} \) |
| 41 | \( 1 + 3.45T + 41T^{2} \) |
| 43 | \( 1 - 7.74T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 59 | \( 1 + 1.21T + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 1.48T + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 4.55T + 89T^{2} \) |
| 97 | \( 1 + 3.08T + 97T^{2} \) |
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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
−8.168590316376121013674599763275, −7.40668977714141858217354255464, −6.46705689779012406439083069260, −6.28599748962958721444077278940, −5.42472293159041942392918582305, −3.99868183845599482612441370199, −2.96798577385461639155200746599, −2.58609856214543965447553127061, −1.53790075781868685541995195830, −0.29719725746046137883314215359,
0.29719725746046137883314215359, 1.53790075781868685541995195830, 2.58609856214543965447553127061, 2.96798577385461639155200746599, 3.99868183845599482612441370199, 5.42472293159041942392918582305, 6.28599748962958721444077278940, 6.46705689779012406439083069260, 7.40668977714141858217354255464, 8.168590316376121013674599763275