L(s) = 1 | + (0.5 + 0.866i)3-s + (−1.60 + 2.77i)5-s + (−1.02 − 2.43i)7-s + (−0.499 + 0.866i)9-s + (2.12 + 3.68i)11-s + 3.15·13-s − 3.20·15-s + (2.20 + 3.81i)17-s + (1.57 − 2.73i)19-s + (1.60 − 2.10i)21-s + (−2.20 + 3.81i)23-s + (−2.62 − 4.54i)25-s − 0.999·27-s − 7.20·29-s + (−1.02 − 1.77i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.715 + 1.23i)5-s + (−0.387 − 0.922i)7-s + (−0.166 + 0.288i)9-s + (0.640 + 1.10i)11-s + 0.874·13-s − 0.826·15-s + (0.533 + 0.924i)17-s + (0.361 − 0.626i)19-s + (0.349 − 0.459i)21-s + (−0.459 + 0.795i)23-s + (−0.524 − 0.909i)25-s − 0.192·27-s − 1.33·29-s + (−0.183 − 0.318i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.187042434\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.187042434\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.02 + 2.43i)T \) |
good | 5 | \( 1 + (1.60 - 2.77i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.12 - 3.68i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 + (-2.20 - 3.81i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.57 + 2.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.20 - 3.81i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.20T + 29T^{2} \) |
| 31 | \( 1 + (1.02 + 1.77i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.82 - 8.35i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 0.750T + 43T^{2} \) |
| 47 | \( 1 + (-1.20 + 2.08i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.64 + 2.85i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.12 - 7.14i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.04 - 7.01i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.57 - 4.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.40T + 71T^{2} \) |
| 73 | \( 1 + (7.62 + 13.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.22 + 14.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + (1.20 - 2.08i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−10.13415200233085928746640487178, −9.283799602861046951262894638988, −8.225085989875997618916877547019, −7.34463173324986953562543106939, −6.92314319886762278871997425661, −5.94597434121113239598304846750, −4.57125505799470536701346441347, −3.60522885662404922096036149103, −3.39258147283734611911568689031, −1.71285743430930006121464394363,
0.48076541119475748237580746525, 1.70447062896640533366404696549, 3.23499032839407748707885266557, 3.89538515294377973594192547882, 5.25925001245738222196948468598, 5.84177866557102741808066684879, 6.83783642898969625244640908604, 7.938487484754486883011801143501, 8.565404945134111081431049623564, 8.966093554980106860033931362889