L(s) = 1 | − 2.41·2-s − 2.29·3-s + 3.85·4-s + 1.32·5-s + 5.55·6-s + 0.993·7-s − 4.49·8-s + 2.26·9-s − 3.20·10-s + 2.10·11-s − 8.85·12-s + 3.07·13-s − 2.40·14-s − 3.03·15-s + 3.15·16-s + 6.60·17-s − 5.48·18-s − 0.0631·19-s + 5.10·20-s − 2.28·21-s − 5.09·22-s − 5.94·23-s + 10.3·24-s − 3.25·25-s − 7.45·26-s + 1.67·27-s + 3.83·28-s + ⋯ |
L(s) = 1 | − 1.71·2-s − 1.32·3-s + 1.92·4-s + 0.591·5-s + 2.26·6-s + 0.375·7-s − 1.58·8-s + 0.756·9-s − 1.01·10-s + 0.634·11-s − 2.55·12-s + 0.853·13-s − 0.642·14-s − 0.783·15-s + 0.789·16-s + 1.60·17-s − 1.29·18-s − 0.0144·19-s + 1.14·20-s − 0.497·21-s − 1.08·22-s − 1.23·23-s + 2.10·24-s − 0.650·25-s − 1.46·26-s + 0.323·27-s + 0.724·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 - 1.32T + 5T^{2} \) |
| 7 | \( 1 - 0.993T + 7T^{2} \) |
| 11 | \( 1 - 2.10T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 - 6.60T + 17T^{2} \) |
| 19 | \( 1 + 0.0631T + 19T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 31 | \( 1 - 5.42T + 31T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 - 0.835T + 43T^{2} \) |
| 47 | \( 1 + 0.379T + 47T^{2} \) |
| 53 | \( 1 + 2.25T + 53T^{2} \) |
| 59 | \( 1 + 7.33T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 + 8.31T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 1.93T + 79T^{2} \) |
| 83 | \( 1 - 7.16T + 83T^{2} \) |
| 89 | \( 1 + 3.20T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−8.152861108817099477737352967783, −7.49818063653595384008946968921, −6.57287046680055360140732169754, −6.06121751067877184074261169833, −5.52744775065208792758188318875, −4.40718102656231644747114077895, −3.15808289014204550114328594725, −1.69350332097963047368611574394, −1.25401336107538663703940123637, 0,
1.25401336107538663703940123637, 1.69350332097963047368611574394, 3.15808289014204550114328594725, 4.40718102656231644747114077895, 5.52744775065208792758188318875, 6.06121751067877184074261169833, 6.57287046680055360140732169754, 7.49818063653595384008946968921, 8.152861108817099477737352967783