L(s) = 1 | − 2-s + (0.933 − 1.45i)3-s + 4-s + (−0.296 + 0.514i)5-s + (−0.933 + 1.45i)6-s + (2.32 − 1.26i)7-s − 8-s + (−1.25 − 2.72i)9-s + (0.296 − 0.514i)10-s + (0.296 + 0.514i)11-s + (0.933 − 1.45i)12-s + (−1.25 − 2.17i)13-s + (−2.32 + 1.26i)14-s + (0.472 + 0.912i)15-s + 16-s + (1.46 − 2.52i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.538 − 0.842i)3-s + 0.5·4-s + (−0.132 + 0.229i)5-s + (−0.381 + 0.595i)6-s + (0.878 − 0.478i)7-s − 0.353·8-s + (−0.419 − 0.907i)9-s + (0.0938 − 0.162i)10-s + (0.0894 + 0.154i)11-s + (0.269 − 0.421i)12-s + (−0.348 − 0.603i)13-s + (−0.621 + 0.338i)14-s + (0.122 + 0.235i)15-s + 0.250·16-s + (0.354 − 0.613i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.863064 - 0.405157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.863064 - 0.405157i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.933 + 1.45i)T \) |
| 7 | \( 1 + (-2.32 + 1.26i)T \) |
good | 5 | \( 1 + (0.296 - 0.514i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.296 - 0.514i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.25 + 2.17i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.46 + 2.52i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.23 - 3.86i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.09 - 5.36i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.136 + 0.236i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.58 - 9.66i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + (-4.02 + 6.97i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 8.64T + 59T^{2} \) |
| 61 | \( 1 + 6.64T + 61T^{2} \) |
| 67 | \( 1 + 1.91T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + (-3.95 + 6.85i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 9.24T + 79T^{2} \) |
| 83 | \( 1 + (-3.85 + 6.66i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.21 + 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.86 + 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.23214841466398813589838160687, −12.09272900853330935809398248859, −11.22750559284703742612475185952, −9.996672597497212720149546362732, −8.849693736931540421121569971305, −7.62045858941007342609328917881, −7.31175292918449682689485407400, −5.56369873423253165324445189779, −3.36667418652079540664379115619, −1.55177684972531063012113657708,
2.33583682642769888899118225151, 4.20953944430714044671990300953, 5.54977942797210003983703452427, 7.39327235666563156657912512035, 8.519243602052107907506029404002, 9.123207967030880575190222058389, 10.30791340036603909834077817022, 11.24776152228810882833312941281, 12.20719165931778426288293864829, 13.79669832597243754337526519283