L(s) = 1 | + (1.20 − 2.08i)2-s + (0.448 + 0.776i)3-s + (−1.90 − 3.29i)4-s + (−0.448 + 0.776i)5-s + 2.15·6-s + (0.117 − 2.64i)7-s − 4.34·8-s + (1.09 − 1.90i)9-s + (1.07 + 1.87i)10-s + (−0.652 − 1.13i)11-s + (1.70 − 2.95i)12-s + 0.896·13-s + (−5.37 − 3.42i)14-s − 0.803·15-s + (−1.42 + 2.47i)16-s + (1.32 + 2.30i)17-s + ⋯ |
L(s) = 1 | + (0.851 − 1.47i)2-s + (0.258 + 0.448i)3-s + (−0.950 − 1.64i)4-s + (−0.200 + 0.347i)5-s + 0.881·6-s + (0.0442 − 0.999i)7-s − 1.53·8-s + (0.366 − 0.634i)9-s + (0.341 + 0.591i)10-s + (−0.196 − 0.340i)11-s + (0.492 − 0.852i)12-s + 0.248·13-s + (−1.43 − 0.916i)14-s − 0.207·15-s + (−0.356 + 0.618i)16-s + (0.322 + 0.558i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01095 - 1.71307i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01095 - 1.71307i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.117 + 2.64i)T \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + (-1.20 + 2.08i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.448 - 0.776i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.448 - 0.776i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.652 + 1.13i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.896T + 13T^{2} \) |
| 17 | \( 1 + (-1.32 - 2.30i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.50 - 2.60i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.67 - 4.63i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.155T + 29T^{2} \) |
| 31 | \( 1 + (-4.25 - 7.37i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.03 + 1.78i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + 4.43T + 43T^{2} \) |
| 47 | \( 1 + (1.12 - 1.95i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.03 - 8.72i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.91 + 5.05i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.33 + 2.30i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.10 + 3.65i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.45T + 71T^{2} \) |
| 73 | \( 1 + (7.00 + 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.12 + 12.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.73T + 83T^{2} \) |
| 89 | \( 1 + (2.15 - 3.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−11.51063805116190537977317921474, −10.50516683065059351108983160631, −10.22185946381664717433584412366, −9.088947229481930122664343534941, −7.66567997385056122722216736290, −6.27339494256712566746177615080, −4.84540649761199154129730572470, −3.76854643482262177768681129187, −3.28278340169382457419401472611, −1.37888708210959600307973133890,
2.52428238375729260954924751300, 4.35337305251035312706967162320, 5.13201653354661508014941887059, 6.22640214306964713274233262222, 7.13042313393216246534999659793, 8.166320436004057169019419173800, 8.606014418868341535232957751991, 10.06127202525298375949692455664, 11.64205882109706027467180944778, 12.57753142271455949635771884951