L(s) = 1 | − 2.96i·2-s − 4.77·4-s − 3.25·5-s + 11.8·7-s + 2.31i·8-s + 9.63i·10-s + 16.1i·11-s + 14.1i·13-s − 35.0i·14-s − 12.2·16-s + 32.0·17-s + 14.7·19-s + 15.5·20-s + 47.8·22-s + 14.5i·23-s + ⋯ |
L(s) = 1 | − 1.48i·2-s − 1.19·4-s − 0.650·5-s + 1.68·7-s + 0.288i·8-s + 0.963i·10-s + 1.46i·11-s + 1.08i·13-s − 2.50i·14-s − 0.767·16-s + 1.88·17-s + 0.778·19-s + 0.776·20-s + 2.17·22-s + 0.634i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.901052398\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.901052398\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + (-18.5 - 55.9i)T \) |
good | 2 | \( 1 + 2.96iT - 4T^{2} \) |
| 5 | \( 1 + 3.25T + 25T^{2} \) |
| 7 | \( 1 - 11.8T + 49T^{2} \) |
| 11 | \( 1 - 16.1iT - 121T^{2} \) |
| 13 | \( 1 - 14.1iT - 169T^{2} \) |
| 17 | \( 1 - 32.0T + 289T^{2} \) |
| 19 | \( 1 - 14.7T + 361T^{2} \) |
| 23 | \( 1 - 14.5iT - 529T^{2} \) |
| 29 | \( 1 - 12.7T + 841T^{2} \) |
| 31 | \( 1 - 27.0iT - 961T^{2} \) |
| 37 | \( 1 + 26.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 37.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 51.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 79.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 86.2T + 2.80e3T^{2} \) |
| 61 | \( 1 - 92.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 88.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 13.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 42.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 1.74T + 6.24e3T^{2} \) |
| 83 | \( 1 + 12.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 69.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 101. iT - 9.40e3T^{2} \) |
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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
−10.59079709118800715984220013813, −9.850331479357134435985363783561, −8.963998428149575572984023982894, −7.76616453477374925549855419103, −7.22427021462139098604387163133, −5.27238897590597687592884892480, −4.44989264536817601471003283801, −3.59507560428721399564401762027, −2.06342437203552472704920082395, −1.27888766731640116567958697603,
0.931258543661334938697959455721, 3.14567845365554669374591125309, 4.55681206173944911570476761162, 5.45646087346094835929331690749, 6.05538081587188635832232107227, 7.60044639141277016648751768246, 8.025125331816768790524119042319, 8.294781318356221410902432125238, 9.695739608245525238288765050472, 11.07769328477216966043854411875