L(s) = 1 | + i·3-s + (1.48 − 1.67i)5-s − i·7-s − 9-s + 0.387·11-s − 2.96i·13-s + (1.67 + 1.48i)15-s − 3.35i·17-s + 2.96·19-s + 21-s − 0.962i·23-s + (−0.612 − 4.96i)25-s − i·27-s − 1.22·29-s + 2.96·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.662 − 0.749i)5-s − 0.377i·7-s − 0.333·9-s + 0.116·11-s − 0.821i·13-s + (0.432 + 0.382i)15-s − 0.812i·17-s + 0.679·19-s + 0.218·21-s − 0.200i·23-s + (−0.122 − 0.992i)25-s − 0.192i·27-s − 0.227·29-s + 0.532·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56551 - 0.592870i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56551 - 0.592870i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.48 + 1.67i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 0.387T + 11T^{2} \) |
| 13 | \( 1 + 2.96iT - 13T^{2} \) |
| 17 | \( 1 + 3.35iT - 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 + 0.962iT - 23T^{2} \) |
| 29 | \( 1 + 1.22T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + 5.92iT - 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 - 10.7iT - 43T^{2} \) |
| 47 | \( 1 + 3.22iT - 47T^{2} \) |
| 53 | \( 1 + 5.66iT - 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 - 6.18iT - 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 + 3.73T + 89T^{2} \) |
| 97 | \( 1 - 7.73iT - 97T^{2} \) |
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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
−9.885778969627113954682435063023, −9.489613458462210739808213628626, −8.515150387801085087034316164631, −7.69163639906254334406665696172, −6.52343680098551348525983667549, −5.47314906069969668432265850019, −4.88107921811860109683108755417, −3.75888069764170828446791524790, −2.53106824840274689586124317191, −0.880136919863549723345630391883,
1.57917781156816028335870646110, 2.58453191598483007313536815465, 3.74565359163872215119227530385, 5.19194085289017240617980738857, 6.11733935312212161875801598594, 6.75272310290887359028063369365, 7.62221222468268609972321000620, 8.646722361872603709779842485508, 9.466253876440716938492076277820, 10.27313282699217597963898637244