L(s) = 1 | − 1.41·2-s + 2.00·4-s − 2.23i·5-s + 1.16i·7-s − 2.82·8-s + 3.16i·10-s − 10.6i·11-s − 15.0·13-s − 1.65i·14-s + 4.00·16-s − 20.0i·17-s + 22.5i·19-s − 4.47i·20-s + 15.0i·22-s + (20.9 − 9.45i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.447i·5-s + 0.167i·7-s − 0.353·8-s + 0.316i·10-s − 0.964i·11-s − 1.16·13-s − 0.118i·14-s + 0.250·16-s − 1.18i·17-s + 1.18i·19-s − 0.223i·20-s + 0.682i·22-s + (0.911 − 0.410i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2811221019\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2811221019\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-20.9 + 9.45i)T \) |
good | 7 | \( 1 - 1.16iT - 49T^{2} \) |
| 11 | \( 1 + 10.6iT - 121T^{2} \) |
| 13 | \( 1 + 15.0T + 169T^{2} \) |
| 17 | \( 1 + 20.0iT - 289T^{2} \) |
| 19 | \( 1 - 22.5iT - 361T^{2} \) |
| 29 | \( 1 + 32.5T + 841T^{2} \) |
| 31 | \( 1 + 27.0T + 961T^{2} \) |
| 37 | \( 1 + 53.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 9.43T + 1.68e3T^{2} \) |
| 43 | \( 1 - 36.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 49.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 104. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 53.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 23.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 59.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 55.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 8.77T + 5.32e3T^{2} \) |
| 79 | \( 1 - 57.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 55.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 19.8iT - 9.40e3T^{2} \) |
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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
−9.183392581366552656664691067461, −8.575665687435946628741436642961, −7.52710061257061250853628354349, −7.24216899817165884753763154382, −5.91018537956020463585384389045, −5.42739572477726405974362829567, −4.31573698492478643306219538979, −3.16163575212058713285272320188, −2.24471580913846047295336817238, −0.978900672777012873843105186459,
0.10080828872987075120054204903, 1.65845132001349417509796288882, 2.50346822657799296290558502037, 3.58075212966513391509425608064, 4.68436682595767669962698215912, 5.54065238254794892263738186672, 6.70553795634574546812194530171, 7.18511941896127900080375526252, 7.78001742884188955003418521027, 8.843827919691169135505756418150