L(s) = 1 | + (0.219 − 0.126i)3-s + (−0.5 + 0.866i)5-s + (−0.978 + 2.45i)7-s + (−1.46 + 2.54i)9-s + (−1.81 − 3.14i)11-s + 5.36·13-s + 0.253i·15-s + (−4.46 + 2.58i)17-s + (−5.49 − 3.17i)19-s + (0.0966 + 0.663i)21-s + (−0.231 − 0.133i)23-s + (−0.499 − 0.866i)25-s + 1.50i·27-s + 2.99i·29-s + (−2.72 − 4.71i)31-s + ⋯ |
L(s) = 1 | + (0.126 − 0.0731i)3-s + (−0.223 + 0.387i)5-s + (−0.369 + 0.929i)7-s + (−0.489 + 0.847i)9-s + (−0.547 − 0.947i)11-s + 1.48·13-s + 0.0654i·15-s + (−1.08 + 0.625i)17-s + (−1.25 − 0.727i)19-s + (0.0210 + 0.144i)21-s + (−0.0482 − 0.0278i)23-s + (−0.0999 − 0.173i)25-s + 0.289i·27-s + 0.556i·29-s + (−0.489 − 0.847i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4517664308\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4517664308\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.978 - 2.45i)T \) |
good | 3 | \( 1 + (-0.219 + 0.126i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.81 + 3.14i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 17 | \( 1 + (4.46 - 2.58i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.49 + 3.17i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.231 + 0.133i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.99iT - 29T^{2} \) |
| 31 | \( 1 + (2.72 + 4.71i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.48 + 4.32i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 5.33T + 43T^{2} \) |
| 47 | \( 1 + (2.26 - 3.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.09 + 1.78i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.83 - 3.94i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.63 - 4.55i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.963 + 1.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15.9iT - 71T^{2} \) |
| 73 | \( 1 + (-7.12 + 4.11i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.94 - 5.74i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.75iT - 83T^{2} \) |
| 89 | \( 1 + (-4.77 - 2.75i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.98iT - 97T^{2} \) |
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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
−10.54007843057122085108401239100, −9.058874630110726649939195967496, −8.587903729011905379860328146807, −8.037480624041093924700392775878, −6.69291169071800077685171232724, −6.06850210641335599827673693682, −5.25052205877607035530690656082, −3.95697172252962421169985151745, −2.94226496808431836223613977362, −2.03929956833369617693483300772,
0.18249132083000499614545577890, 1.78232224732527122952446883697, 3.34737684524348458362011566297, 4.05243779953076337688379105051, 4.98835876210076631380711009395, 6.30421230591917410027711300963, 6.78413343295817416914567784109, 7.904996907742780106453419456593, 8.675103176255303731491884727949, 9.340748360017808157338414853940