L(s) = 1 | + (−0.623 − 1.07i)2-s + (−0.5 − 0.866i)3-s + (0.222 − 0.385i)4-s − 2.80·5-s + (−0.623 + 1.07i)6-s + (2.40 − 4.15i)7-s − 3.04·8-s + (−0.499 + 0.866i)9-s + (1.74 + 3.02i)10-s + (−0.733 − 1.27i)11-s − 0.445·12-s − 5.98·14-s + (1.40 + 2.42i)15-s + (1.45 + 2.52i)16-s + (1.22 − 2.11i)17-s + 1.24·18-s + ⋯ |
L(s) = 1 | + (−0.440 − 0.763i)2-s + (−0.288 − 0.499i)3-s + (0.111 − 0.192i)4-s − 1.25·5-s + (−0.254 + 0.440i)6-s + (0.907 − 1.57i)7-s − 1.07·8-s + (−0.166 + 0.288i)9-s + (0.552 + 0.956i)10-s + (−0.221 − 0.383i)11-s − 0.128·12-s − 1.60·14-s + (0.361 + 0.626i)15-s + (0.363 + 0.630i)16-s + (0.296 − 0.513i)17-s + 0.293·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.244661 + 0.498394i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.244661 + 0.498394i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.623 + 1.07i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 2.80T + 5T^{2} \) |
| 7 | \( 1 + (-2.40 + 4.15i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.733 + 1.27i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.22 + 2.11i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.27 - 2.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.75 - 3.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.925 + 1.60i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 + (-2.27 - 3.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.623 + 1.07i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.19 - 2.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 8.85T + 53T^{2} \) |
| 59 | \( 1 + (1.08 - 1.88i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.91 + 6.78i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.79 - 3.10i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.41 + 7.65i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 + 4.02T + 79T^{2} \) |
| 83 | \( 1 + 0.652T + 83T^{2} \) |
| 89 | \( 1 + (3.14 + 5.45i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.01 + 8.68i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.75948152988387882526207236814, −9.702665482258327620211893590240, −8.398855880685278558323141172400, −7.64025696786318458345553413150, −7.00851883699688111577178816191, −5.60509209746762363061836574989, −4.32972556674601684195301237049, −3.30355837569467803352091487550, −1.54846859320585799697169659112, −0.40335543681034526766362535944,
2.52194600600372058720521402482, 3.87260986627731040037512834077, 5.06781234843590150447712330282, 5.93519573568424868358637514578, 7.16508200994173259069433388491, 7.973100353451714596104324609009, 8.668423377220578874580136551163, 9.267375663581040744267204723877, 10.82349032482150823382869775954, 11.46172124883636306979723020944