L(s) = 1 | + (−0.705 + 0.512i)2-s + (0.309 + 0.951i)3-s + (−0.383 + 1.17i)4-s + (3.28 + 2.38i)5-s + (−0.705 − 0.512i)6-s + (−0.309 + 0.951i)7-s + (−0.872 − 2.68i)8-s + (−0.809 + 0.587i)9-s − 3.54·10-s + (−2.96 − 1.48i)11-s − 1.24·12-s + (4.92 − 3.57i)13-s + (−0.269 − 0.828i)14-s + (−1.25 + 3.86i)15-s + (−0.0153 − 0.0111i)16-s + (−1.37 − 1.00i)17-s + ⋯ |
L(s) = 1 | + (−0.498 + 0.362i)2-s + (0.178 + 0.549i)3-s + (−0.191 + 0.589i)4-s + (1.47 + 1.06i)5-s + (−0.287 − 0.209i)6-s + (−0.116 + 0.359i)7-s + (−0.308 − 0.949i)8-s + (−0.269 + 0.195i)9-s − 1.12·10-s + (−0.894 − 0.446i)11-s − 0.358·12-s + (1.36 − 0.992i)13-s + (−0.0719 − 0.221i)14-s + (−0.324 + 0.998i)15-s + (−0.00383 − 0.00278i)16-s + (−0.333 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.598918 + 0.960476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.598918 + 0.960476i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (2.96 + 1.48i)T \) |
good | 2 | \( 1 + (0.705 - 0.512i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (-3.28 - 2.38i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (-4.92 + 3.57i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.37 + 1.00i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.68 - 5.19i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 6.39T + 23T^{2} \) |
| 29 | \( 1 + (-1.46 + 4.51i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.52 + 2.56i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.753 + 2.32i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.992 + 3.05i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 0.127T + 43T^{2} \) |
| 47 | \( 1 + (-2.61 - 8.05i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.81 + 2.77i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.43 + 4.40i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.29 + 3.12i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + (1.79 + 1.30i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.72 - 5.30i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.07 - 3.68i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.103 - 0.0750i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 8.12T + 89T^{2} \) |
| 97 | \( 1 + (-2.10 + 1.52i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.73017441654917654121353630173, −11.23340105642225035359659481539, −10.20960931522998654413375552637, −9.770751279085897476189312849949, −8.562390340261654210187603196684, −7.77560236221261264875873557322, −6.25020967997906552649209792705, −5.68922988845287931831337289327, −3.63409622504588684287374530609, −2.59029320353938008005655034576,
1.21863370719623559506071544362, 2.24488964203738058932447451798, 4.67790924537485488534269721373, 5.72579747906606545886666812640, 6.65097802868231613376671496246, 8.405671216667906489765218471553, 8.997653061544666893670398608543, 9.877400409491674110206513168239, 10.66291742198238712117452991555, 11.87389465609368162128052771681