L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 13-s + 16-s + 6·17-s + 4·19-s + 20-s + 25-s + 26-s − 6·29-s + 4·31-s − 32-s − 6·34-s − 10·37-s − 4·38-s − 40-s + 6·41-s + 8·43-s − 50-s − 52-s + 6·53-s + 6·58-s − 12·59-s − 14·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.196·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s − 1.64·37-s − 0.648·38-s − 0.158·40-s + 0.937·41-s + 1.21·43-s − 0.141·50-s − 0.138·52-s + 0.824·53-s + 0.787·58-s − 1.56·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.884176872\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.884176872\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.32348181170331, −13.92350109334275, −13.47849538724694, −12.62767356785222, −12.30647320525934, −11.85260741399686, −11.23892747554515, −10.59773187345517, −10.26742879586816, −9.684631745651272, −9.225689656298729, −8.856522280236919, −8.031599481029687, −7.498772768335837, −7.332232828071617, −6.449873957000038, −5.847953598509866, −5.480573564458334, −4.814023889428378, −3.984432514400027, −3.216689469550975, −2.814891439965096, −1.893257224341741, −1.332612693755464, −0.5548873948317094,
0.5548873948317094, 1.332612693755464, 1.893257224341741, 2.814891439965096, 3.216689469550975, 3.984432514400027, 4.814023889428378, 5.480573564458334, 5.847953598509866, 6.449873957000038, 7.332232828071617, 7.498772768335837, 8.031599481029687, 8.856522280236919, 9.225689656298729, 9.684631745651272, 10.26742879586816, 10.59773187345517, 11.23892747554515, 11.85260741399686, 12.30647320525934, 12.62767356785222, 13.47849538724694, 13.92350109334275, 14.32348181170331