| L(s) = 1 | − 4.27e3i·3-s − 5.92e4·5-s − 1.04e6i·7-s − 1.34e7·9-s − 1.23e7i·11-s + 5.91e7·13-s + 2.53e8i·15-s + 2.67e8·17-s + 6.60e8i·19-s − 4.46e9·21-s − 1.04e8i·23-s − 2.59e9·25-s + 3.71e10i·27-s + 1.09e10·29-s − 9.03e9i·31-s + ⋯ |
| L(s) = 1 | − 1.95i·3-s − 0.758·5-s − 1.26i·7-s − 2.81·9-s − 0.631i·11-s + 0.942·13-s + 1.48i·15-s + 0.650·17-s + 0.738i·19-s − 2.47·21-s − 0.0306i·23-s − 0.424·25-s + 3.55i·27-s + 0.633·29-s − 0.328i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.457660 + 0.792690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.457660 + 0.792690i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 4.27e3iT - 4.78e6T^{2} \) |
| 5 | \( 1 + 5.92e4T + 6.10e9T^{2} \) |
| 7 | \( 1 + 1.04e6iT - 6.78e11T^{2} \) |
| 11 | \( 1 + 1.23e7iT - 3.79e14T^{2} \) |
| 13 | \( 1 - 5.91e7T + 3.93e15T^{2} \) |
| 17 | \( 1 - 2.67e8T + 1.68e17T^{2} \) |
| 19 | \( 1 - 6.60e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 + 1.04e8iT - 1.15e19T^{2} \) |
| 29 | \( 1 - 1.09e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + 9.03e9iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 1.07e11T + 9.01e21T^{2} \) |
| 41 | \( 1 - 2.71e11T + 3.79e22T^{2} \) |
| 43 | \( 1 - 1.56e11iT - 7.38e22T^{2} \) |
| 47 | \( 1 + 6.62e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 1.98e12T + 1.37e24T^{2} \) |
| 59 | \( 1 + 7.45e11iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 4.18e11T + 9.87e24T^{2} \) |
| 67 | \( 1 + 3.33e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 + 8.61e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 1.63e13T + 1.22e26T^{2} \) |
| 79 | \( 1 + 2.19e13iT - 3.68e26T^{2} \) |
| 83 | \( 1 - 1.49e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 + 1.22e13T + 1.95e27T^{2} \) |
| 97 | \( 1 + 2.31e13T + 6.52e27T^{2} \) |
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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
−14.21209258520559545883152377391, −13.41024042184468458108642220477, −12.14833914727498103558206817781, −10.99161325938826647377532661251, −8.267987962335364485487411124730, −7.44283707668746807684508058381, −6.13893938342966794431321113816, −3.45355918093293927699225143285, −1.37141060323344517318940352315, −0.35680627879401868361208211450,
2.99555730261388886154641435701, 4.33865198919160258519561371351, 5.65439992470305527846048369084, 8.448335467308203625550200473171, 9.489952441810520366919434483927, 10.92370624039253234251768414750, 12.02701231907971971929249659622, 14.45081366088008752024531739831, 15.64787358479229150246646712620, 15.86090516321213839596879044361
