L(s) = 1 | + (−0.5 − 0.866i)3-s + (2.12 + 1.22i)5-s + (−0.499 + 0.866i)9-s + (2.12 − 1.22i)11-s − 4.89i·13-s − 2.44i·15-s + (−2.12 + 1.22i)17-s + (1 − 1.73i)19-s + (−6.36 − 3.67i)23-s + (0.499 + 0.866i)25-s + 0.999·27-s + 6·29-s + (−4 − 6.92i)31-s + (−2.12 − 1.22i)33-s + (−2 + 3.46i)37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.948 + 0.547i)5-s + (−0.166 + 0.288i)9-s + (0.639 − 0.369i)11-s − 1.35i·13-s − 0.632i·15-s + (−0.514 + 0.297i)17-s + (0.229 − 0.397i)19-s + (−1.32 − 0.766i)23-s + (0.0999 + 0.173i)25-s + 0.192·27-s + 1.11·29-s + (−0.718 − 1.24i)31-s + (−0.369 − 0.213i)33-s + (−0.328 + 0.569i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.681585250\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.681585250\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + (-2.12 - 1.22i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.12 + 1.22i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + (2.12 - 1.22i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.36 + 3.67i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.34iT - 41T^{2} \) |
| 43 | \( 1 + 4.89iT - 43T^{2} \) |
| 47 | \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.2iT - 71T^{2} \) |
| 73 | \( 1 + (-12.7 + 7.34i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.48 - 4.89i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (10.6 + 6.12i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−8.664112266654065585304861339808, −8.076321600256402149060356099314, −7.12363824469131044800370076645, −6.27248433950940626256940116751, −5.97130361136082310649151070030, −5.01982271692768564683111061404, −3.84839612273862822053438899270, −2.74419680636911592601699580804, −1.96622288449934021402527546144, −0.59667187597707975811514369782,
1.38036711545011244182255370376, 2.19878250973835774707589691550, 3.66834958311122374618159206961, 4.42482339946714923429809616059, 5.20323457421702508154381541519, 6.02142150210284893618772497087, 6.66808759157727447001310661618, 7.56780574776963909632512242623, 8.783605501369434309630028180464, 9.221153493590878159676793868614