L(s) = 1 | + 10.4i·2-s − 77.1·4-s + 72.3·5-s − 9.31i·7-s − 471. i·8-s + 755. i·10-s + 308.·11-s + 245.·13-s + 97.3·14-s + 2.46e3·16-s − 762.·17-s + 2.84e3i·19-s − 5.58e3·20-s + 3.22e3i·22-s + (2.02e3 + 1.53e3i)23-s + ⋯ |
L(s) = 1 | + 1.84i·2-s − 2.41·4-s + 1.29·5-s − 0.0718i·7-s − 2.60i·8-s + 2.39i·10-s + 0.769·11-s + 0.403·13-s + 0.132·14-s + 2.40·16-s − 0.639·17-s + 1.81i·19-s − 3.12·20-s + 1.42i·22-s + (0.797 + 0.603i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.960890947\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.960890947\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + (-2.02e3 - 1.53e3i)T \) |
good | 2 | \( 1 - 10.4iT - 32T^{2} \) |
| 5 | \( 1 - 72.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 9.31iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 308.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 245.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 762.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.84e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 6.02e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.44e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.27e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.41e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 7.76e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.42e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.79e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.35e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.94e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 7.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.82e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 7.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.15e4iT - 8.58e9T^{2} \) |
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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
−12.57606495286137805290928144504, −10.77969043025668856432396947882, −9.498732337338499728289780344840, −9.053984457682981218207776114864, −7.85993339595646487418829506236, −6.74365282465328246326214425029, −6.00219522640360810025589574915, −5.21856909953805617883505911255, −3.81179815613945127091655566379, −1.45099853720340358209621065307,
0.63155852647738929902189200712, 1.86821878700273341242917368626, 2.75127211504067553351099818886, 4.17518732958178740802216941249, 5.30118565930222413096887646710, 6.67425041720257511767883541706, 8.784532547273234639415650923352, 9.223337033107889517901168000863, 10.15088204719932501158636189937, 11.02651453508086960199992694342