L(s) = 1 | − i·2-s + (1.72 + 0.144i)3-s − 4-s + 5-s + (0.144 − 1.72i)6-s + i·8-s + (2.95 + 0.498i)9-s − i·10-s + 5.90i·11-s + (−1.72 − 0.144i)12-s + 1.19i·13-s + (1.72 + 0.144i)15-s + 16-s − 0.519·17-s + (0.498 − 2.95i)18-s + 5.46i·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.996 + 0.0833i)3-s − 0.5·4-s + 0.447·5-s + (0.0589 − 0.704i)6-s + 0.353i·8-s + (0.986 + 0.166i)9-s − 0.316i·10-s + 1.78i·11-s + (−0.498 − 0.0416i)12-s + 0.330i·13-s + (0.445 + 0.0372i)15-s + 0.250·16-s − 0.126·17-s + (0.117 − 0.697i)18-s + 1.25i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.467463111\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.467463111\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.72 - 0.144i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5.90iT - 11T^{2} \) |
| 13 | \( 1 - 1.19iT - 13T^{2} \) |
| 17 | \( 1 + 0.519T + 17T^{2} \) |
| 19 | \( 1 - 5.46iT - 19T^{2} \) |
| 23 | \( 1 + 6.81iT - 23T^{2} \) |
| 29 | \( 1 - 6.27iT - 29T^{2} \) |
| 31 | \( 1 - 3.20iT - 31T^{2} \) |
| 37 | \( 1 + 4.15T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 + 5.45T + 47T^{2} \) |
| 53 | \( 1 + 4.12iT - 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 + 9.24iT - 61T^{2} \) |
| 67 | \( 1 + 7.37T + 67T^{2} \) |
| 71 | \( 1 + 9.94iT - 71T^{2} \) |
| 73 | \( 1 + 9.04iT - 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 2.22T + 89T^{2} \) |
| 97 | \( 1 - 9.01iT - 97T^{2} \) |
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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
−9.471910728486365612563068174173, −9.002476993669029464316213947168, −8.046639531572678013503720960042, −7.26034438981927022086882606376, −6.37076396765875472282428766217, −4.93658340991293996724774305257, −4.35348345366804800278037849305, −3.32507167035768565545412180181, −2.22973544418522651858196483843, −1.60972529477865978097493606580,
0.931397818430243312520378494479, 2.53943499740423801201217966767, 3.40696446284714498931836713166, 4.39559835898561197185668951342, 5.59693210206245029225352570436, 6.18191487567886173626717798728, 7.26167706415032854697036346304, 7.87865337076909026540802519105, 8.745441060400012644223622799898, 9.175312417100415732315792328281