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
−11.87389465609368162128052771681, −10.66291742198238712117452991555, −9.877400409491674110206513168239, −8.997653061544666893670398608543, −8.405671216667906489765218471553, −6.65097802868231613376671496246, −5.72579747906606545886666812640, −4.67790924537485488534269721373, −2.24488964203738058932447451798, −1.21863370719623559506071544362,
2.59029320353938008005655034576, 3.63409622504588684287374530609, 5.68922988845287931831337289327, 6.25020967997906552649209792705, 7.77560236221261264875873557322, 8.562390340261654210187603196684, 9.770751279085897476189312849949, 10.20960931522998654413375552637, 11.23340105642225035359659481539, 12.73017441654917654121353630173