L(s) = 1 | − i·2-s − 4-s + i·7-s + i·8-s + (1 + i)11-s + 14-s + 16-s + (1 − i)22-s + i·25-s − i·28-s + (1 − i)29-s − i·32-s + (−1 − i)37-s + (1 + i)43-s + (−1 − i)44-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + i·7-s + i·8-s + (1 + i)11-s + 14-s + 16-s + (1 − i)22-s + i·25-s − i·28-s + (1 − i)29-s − i·32-s + (−1 − i)37-s + (1 + i)43-s + (−1 − i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9473125618\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9473125618\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-1 - i)T + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (-1 + i)T - iT^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.967813681169342208175029349838, −9.437161047326423074733404257214, −8.771656860453085751454923820629, −7.85035667573313045575570778239, −6.66834373646425384122919679296, −5.61201941496033626014267506117, −4.70368051167754649848998230011, −3.76672239131288494893803170190, −2.59611604592398391046324998051, −1.60613489739121172351148783062,
1.05803658760874879022386712685, 3.30585615892353791783724864143, 4.14426234420912151412183513908, 5.05738762063776045952704665487, 6.24167129196224017674919038079, 6.72230448940592274129180487575, 7.65611866244347104797648494049, 8.506134317780769892805558816464, 9.127131572802656803559456188956, 10.16528615582161488930150417818