L(s) = 1 | + (0.258 − 0.965i)2-s + (−1.33 − 1.10i)3-s + (0.866 + 0.5i)4-s + (−0.151 + 0.565i)5-s + (−1.41 + i)6-s + (−1.15 − 2.38i)7-s + (2.12 − 2.12i)8-s + (0.548 + 2.94i)9-s + (0.507 + 0.292i)10-s + (−1.50 − 5.62i)11-s + (−0.599 − 1.62i)12-s + (2 − 3i)13-s + (−2.59 + 0.500i)14-s + (0.828 − 0.585i)15-s + (−0.500 − 0.866i)16-s + (−0.914 + 1.58i)17-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.769 − 0.639i)3-s + (0.433 + 0.250i)4-s + (−0.0678 + 0.253i)5-s + (−0.577 + 0.408i)6-s + (−0.436 − 0.899i)7-s + (0.749 − 0.749i)8-s + (0.182 + 0.983i)9-s + (0.160 + 0.0926i)10-s + (−0.454 − 1.69i)11-s + (−0.173 − 0.469i)12-s + (0.554 − 0.832i)13-s + (−0.694 + 0.133i)14-s + (0.213 − 0.151i)15-s + (−0.125 − 0.216i)16-s + (−0.221 + 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.534455 - 1.01193i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.534455 - 1.01193i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.33 + 1.10i)T \) |
| 7 | \( 1 + (1.15 + 2.38i)T \) |
| 13 | \( 1 + (-2 + 3i)T \) |
good | 2 | \( 1 + (-0.258 + 0.965i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (0.151 - 0.565i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (1.50 + 5.62i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (0.914 - 1.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.43 + 1.72i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.29 - 5.70i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + iT - 29T^{2} \) |
| 31 | \( 1 + (-0.214 - 0.800i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.03 + 3.86i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-4 - 4i)T + 41iT^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + (0.634 + 0.170i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-7.49 - 4.32i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.45 - 9.16i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.49 - 13.0i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.464 - 0.464i)T + 71iT^{2} \) |
| 73 | \( 1 + (-12.3 + 3.32i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (0.707 + 1.22i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.07 - 3.07i)T + 83iT^{2} \) |
| 89 | \( 1 + (12.0 + 3.23i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (7 - 7i)T - 97iT^{2} \) |
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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
−11.23766702484950556789804797229, −10.95225036216407018718480180524, −10.35231769365556893811868877704, −8.517746325553221598231837155389, −7.47754360924964133680409219839, −6.61585066875022942663266684340, −5.65094816313294648914920402245, −3.92769145520331031766332831476, −2.81039936739004820117198318883, −0.936426856354639800014310621549,
2.24932565920703328184973406783, 4.42793277944690722591180650607, 5.11698616955691382173734457301, 6.40143008306052271646721555633, 6.79698709309280513467033679746, 8.395759146556402609500030165148, 9.447302591863960627779129451989, 10.40706106315230865787418930004, 11.21798334540835532558587713997, 12.29643276479824553803319928162