L(s) = 1 | + (−1 − i)5-s + i·13-s + 2i·17-s + i·25-s + (1 − i)37-s + (1 + i)41-s + i·49-s + 2·53-s − 2·61-s + (1 − i)65-s + (1 − i)73-s + (2 − 2i)85-s + (1 − i)89-s + (1 + i)97-s + (1 + i)109-s + ⋯ |
L(s) = 1 | + (−1 − i)5-s + i·13-s + 2i·17-s + i·25-s + (1 − i)37-s + (1 + i)41-s + i·49-s + 2·53-s − 2·61-s + (1 − i)65-s + (1 − i)73-s + (2 − 2i)85-s + (1 − i)89-s + (1 + i)97-s + (1 + i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9583517946\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9583517946\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 5 | \( 1 + (1 + i)T + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - 2T + T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-1 + i)T - iT^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−8.881280032712771804404653911545, −7.905175551103517088204351829065, −7.60879150908012125039689572300, −6.41919343677687892111192996719, −5.86087061692939696009033054717, −4.70647373636620488158144824503, −4.21580925294549505121109840891, −3.57539908031064634315522150312, −2.17349867152280524075897364071, −1.10247418661650765586022074307,
0.66156577215943567323766281094, 2.50902055084983847112073041410, 3.06771659470846232960731048340, 3.89430784817368792590338407325, 4.82838919246430270098992968157, 5.62912378633963076556557223588, 6.60298547932210727145245960736, 7.33639717729686699128713140326, 7.65588509238764072186343001743, 8.518609341742377314456769295243