| L(s) = 1 | − 1.51·3-s − 3.62·5-s − 0.892·7-s − 0.710·9-s + 11-s − 2.07·13-s + 5.48·15-s − 1.84·17-s − 0.668·19-s + 1.35·21-s + 3.04·23-s + 8.12·25-s + 5.61·27-s + 6.10·29-s + 0.0259·31-s − 1.51·33-s + 3.23·35-s + 3.85·37-s + 3.13·39-s + 0.603·41-s + 12.0·43-s + 2.57·45-s − 4.57·47-s − 6.20·49-s + 2.79·51-s + 53-s − 3.62·55-s + ⋯ |
| L(s) = 1 | − 0.873·3-s − 1.62·5-s − 0.337·7-s − 0.236·9-s + 0.301·11-s − 0.574·13-s + 1.41·15-s − 0.447·17-s − 0.153·19-s + 0.294·21-s + 0.634·23-s + 1.62·25-s + 1.08·27-s + 1.13·29-s + 0.00466·31-s − 0.263·33-s + 0.546·35-s + 0.634·37-s + 0.502·39-s + 0.0941·41-s + 1.83·43-s + 0.383·45-s − 0.667·47-s − 0.886·49-s + 0.391·51-s + 0.137·53-s − 0.488·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 1.51T + 3T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 + 0.892T + 7T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 19 | \( 1 + 0.668T + 19T^{2} \) |
| 23 | \( 1 - 3.04T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 - 0.0259T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 - 0.603T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 59 | \( 1 + 3.89T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 9.62T + 73T^{2} \) |
| 79 | \( 1 + 6.00T + 79T^{2} \) |
| 83 | \( 1 + 3.68T + 83T^{2} \) |
| 89 | \( 1 - 5.66T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.895442484947280601374495649413, −7.15828639447899069777712636860, −6.56702249678731071981409668906, −5.80904054017595889607568977844, −4.77156129590621046038994746018, −4.39844033474517976644395888400, −3.39586374147816819723369297194, −2.62905305761675970209061201176, −0.909863567867264072244213546422, 0,
0.909863567867264072244213546422, 2.62905305761675970209061201176, 3.39586374147816819723369297194, 4.39844033474517976644395888400, 4.77156129590621046038994746018, 5.80904054017595889607568977844, 6.56702249678731071981409668906, 7.15828639447899069777712636860, 7.895442484947280601374495649413
