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
\(L(1)\) |
\(\approx\) |
\(0.6619806123\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6619806123\) |
\(L(\frac{3}{2})\) |
|
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
−10.32276885476827009193723066257, −8.790055146683542153416251689435, −8.245540278261495537688621259171, −7.62505853329781535153058076527, −6.47604069920295559932866790624, −6.05058234865205927650140122376, −5.08146535915190510005103818875, −3.59243034714297179850968112741, −2.99438512647300092842150010786, −1.68043099125549943201610001121,
0.28718914142503749830933158643, 1.59346958875694662507637482209, 3.41516182815561816282692922630, 4.31436557211254117510900414118, 4.88403160810428698149771413600, 5.62148081461295690704413086701, 7.14979395477303005866470148551, 7.56895547488076905898707847005, 8.563720549335686376632967474755, 9.248166061231349777122045270085