L(s) = 1 | + i·3-s + (−1.71 + 1.42i)5-s + 1.37i·7-s − 9-s + 3.21·11-s − 1.24i·13-s + (−1.42 − 1.71i)15-s − 5.00i·17-s − 0.139·19-s − 1.37·21-s + 1.01i·23-s + (0.915 − 4.91i)25-s − i·27-s + 8.55·29-s + 1.37·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.769 + 0.639i)5-s + 0.521i·7-s − 0.333·9-s + 0.969·11-s − 0.345i·13-s + (−0.369 − 0.444i)15-s − 1.21i·17-s − 0.0319·19-s − 0.301·21-s + 0.210i·23-s + (0.183 − 0.983i)25-s − 0.192i·27-s + 1.58·29-s + 0.247·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.664843900\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.664843900\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.71 - 1.42i)T \) |
| 67 | \( 1 + iT \) |
good | 7 | \( 1 - 1.37iT - 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 13 | \( 1 + 1.24iT - 13T^{2} \) |
| 17 | \( 1 + 5.00iT - 17T^{2} \) |
| 19 | \( 1 + 0.139T + 19T^{2} \) |
| 23 | \( 1 - 1.01iT - 23T^{2} \) |
| 29 | \( 1 - 8.55T + 29T^{2} \) |
| 31 | \( 1 - 1.37T + 31T^{2} \) |
| 37 | \( 1 - 5.04iT - 37T^{2} \) |
| 41 | \( 1 + 0.647T + 41T^{2} \) |
| 43 | \( 1 + 9.78iT - 43T^{2} \) |
| 47 | \( 1 + 3.18iT - 47T^{2} \) |
| 53 | \( 1 + 2.68iT - 53T^{2} \) |
| 59 | \( 1 - 9.87T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 16.0iT - 73T^{2} \) |
| 79 | \( 1 - 1.22T + 79T^{2} \) |
| 83 | \( 1 - 3.38iT - 83T^{2} \) |
| 89 | \( 1 - 2.10T + 89T^{2} \) |
| 97 | \( 1 + 8.14iT - 97T^{2} \) |
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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
−8.641769315348333006214050038938, −7.84196787224610056269616437726, −6.96489676564764020279479565627, −6.46979266817399932954462721876, −5.45002874298777680090426251025, −4.70698581424501830458597093436, −3.87594584827128015822129630331, −3.14888375816440384478634702150, −2.36393143140803421151036162815, −0.71744161645113035907798860259,
0.818974267167133799677349584157, 1.56425801619460945369702120018, 2.86946278325175702629709034442, 4.05540643671525158437078316621, 4.25717104611919301476183479567, 5.42162702977408062504120041527, 6.34061974590015134083908449079, 6.90351476933184207458807469287, 7.65857811515848208780387675557, 8.434352193371690630344512488075