L(s) = 1 | + (−0.866 − 1.5i)3-s + (2.38 − 1.13i)7-s + (2.63 + 4.56i)11-s + 2.62·13-s + (−0.209 − 0.362i)17-s + (−1.63 + 2.83i)19-s + (−3.77 − 2.59i)21-s + (3.91 − 6.77i)23-s − 5.19·27-s + 4.27·29-s + (−1.63 − 2.83i)31-s + (4.56 − 7.91i)33-s + (4.98 − 8.63i)37-s + (−2.27 − 3.94i)39-s − 3.72·41-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.866i)3-s + (0.902 − 0.429i)7-s + (0.795 + 1.37i)11-s + 0.728·13-s + (−0.0507 − 0.0879i)17-s + (−0.375 + 0.650i)19-s + (−0.823 − 0.566i)21-s + (0.815 − 1.41i)23-s − 1.00·27-s + 0.793·29-s + (−0.294 − 0.509i)31-s + (0.795 − 1.37i)33-s + (0.819 − 1.41i)37-s + (−0.364 − 0.630i)39-s − 0.581·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31050 - 0.770506i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31050 - 0.770506i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.38 + 1.13i)T \) |
good | 3 | \( 1 + (0.866 + 1.5i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.63 - 4.56i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 + (0.209 + 0.362i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.63 - 2.83i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.91 + 6.77i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 + (1.63 + 2.83i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.98 + 8.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.72T + 41T^{2} \) |
| 43 | \( 1 + 2.15T + 43T^{2} \) |
| 47 | \( 1 + (3.25 - 5.63i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.83 - 4.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.63 - 2.83i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.77 + 11.7i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.76 + 3.04i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 + (3.25 + 5.63i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.63 - 6.30i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.40T + 83T^{2} \) |
| 89 | \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−10.46655079931175382395858448431, −9.439711202994429064574662006937, −8.444159187043064353707137507773, −7.50939748208063404432810372787, −6.81086553420221683922241001319, −6.06986121767843975647920051884, −4.75092454294923231241687925444, −3.96224881433107441035677121778, −2.07469347460716453669495629770, −1.06355585062648644404595131153,
1.37418934929991098085460143080, 3.16752585559258338742474332763, 4.23256867405151727780494334325, 5.17911708521264930780621539019, 5.88378195623161211641256070460, 6.97846634316705582967579005715, 8.352322369462669149786541810102, 8.761994205973019387214281993027, 9.821969975305646413602158627860, 10.75248110888651546977575615766