L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (1.90 − 1.10i)5-s + (−2.24 − 1.40i)7-s + 0.999i·8-s + (−1.10 + 1.90i)10-s + (1.82 + 2.77i)11-s − 1.45·13-s + (2.64 + 0.0990i)14-s + (−0.5 − 0.866i)16-s + (−3.80 + 6.58i)17-s + (0.0903 + 0.156i)19-s − 2.20i·20-s + (−2.96 − 1.49i)22-s + (1.14 + 1.98i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.853 − 0.492i)5-s + (−0.846 − 0.532i)7-s + 0.353i·8-s + (−0.348 + 0.603i)10-s + (0.548 + 0.835i)11-s − 0.404·13-s + (0.706 + 0.0264i)14-s + (−0.125 − 0.216i)16-s + (−0.922 + 1.59i)17-s + (0.0207 + 0.0358i)19-s − 0.492i·20-s + (−0.631 − 0.317i)22-s + (0.238 + 0.413i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.132142283\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.132142283\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.24 + 1.40i)T \) |
| 11 | \( 1 + (-1.82 - 2.77i)T \) |
good | 5 | \( 1 + (-1.90 + 1.10i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 + (3.80 - 6.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0903 - 0.156i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.14 - 1.98i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.45iT - 29T^{2} \) |
| 31 | \( 1 + (-7.40 - 4.27i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.754 + 1.30i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 + 1.58iT - 43T^{2} \) |
| 47 | \( 1 + (-0.472 + 0.272i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.41 + 4.18i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.36 - 3.09i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.86 - 4.95i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.49 + 2.58i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.57T + 71T^{2} \) |
| 73 | \( 1 + (-4.83 + 8.38i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.99 + 3.45i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.84T + 83T^{2} \) |
| 89 | \( 1 + (13.9 - 8.05i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 11.3iT - 97T^{2} \) |
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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
−9.610990720713342680101187161994, −9.079637959850135277442319022483, −8.248140439415034138883370599334, −7.13778406258809915698090073696, −6.58863710987081323889212300936, −5.81261452082849726405045792295, −4.77162094178769501852009093802, −3.75836749031614580906455777252, −2.26534041365644484623916833939, −1.23347066769866457771505312825,
0.60983980846855888122517766971, 2.44742307739792345467066988421, 2.75327109870108473406458618226, 4.11601144390799196361709150549, 5.42082678954976875914863659627, 6.42359959303234924483722180301, 6.74478818842830600225085787471, 7.958281428551174701125891457310, 8.895832358140872804972671659594, 9.557693200590790638073373274574