| L(s) = 1 | − 2.23i·3-s + 2.82·7-s − 2.00·9-s − 2.23i·11-s − 6.32i·13-s − 5·17-s + 2.23i·19-s − 6.32i·21-s − 5.65·23-s − 2.23i·27-s + 6.32i·29-s − 5.00·33-s − 6.32i·37-s − 14.1·39-s + 3·41-s + ⋯ |
| L(s) = 1 | − 1.29i·3-s + 1.06·7-s − 0.666·9-s − 0.674i·11-s − 1.75i·13-s − 1.21·17-s + 0.512i·19-s − 1.38i·21-s − 1.17·23-s − 0.430i·27-s + 1.17i·29-s − 0.870·33-s − 1.03i·37-s − 2.26·39-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440759362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440759362\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.23iT - 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 6.32iT - 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 2.23iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 6.32iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6.32iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 8.94iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 12.6iT - 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 6.32iT - 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 15T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 6.70iT - 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.182409233080395365402678022842, −7.68476893211454670393864768260, −6.89156174630178920572360164240, −6.02952971619918590588048418053, −5.46086335597747659639396158472, −4.47781401557369997740620382317, −3.35468623027017796129939924176, −2.30942540519664211852646965614, −1.50046672825610373103566213796, −0.42514818077839116594035130263,
1.69436094035680849385483531906, 2.48331017397380934195413382723, 4.04459667395094798749733183381, 4.34654826700357910885351162280, 4.83670540501466096694593356625, 5.90812439242481346525525789960, 6.81769659504586374408423615752, 7.56855317614927768013451128151, 8.552258364643605042829625578567, 9.119228341992280185056460404321
