| L(s) = 1 | − 2i·7-s + 3·11-s − 4i·13-s − 6i·17-s + 7·19-s + 6i·23-s − 3·29-s + 5·31-s + 4i·37-s + 3·41-s + 8i·43-s + 3·49-s − 6i·53-s + 3·59-s + 14·61-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s + 0.904·11-s − 1.10i·13-s − 1.45i·17-s + 1.60·19-s + 1.25i·23-s − 0.557·29-s + 0.898·31-s + 0.657i·37-s + 0.468·41-s + 1.21i·43-s + 0.428·49-s − 0.824i·53-s + 0.390·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.353962189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.353962189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−7.62070107304344128717546494630, −7.12932128688980414257950114975, −6.43002531206884523582314177981, −5.43906955607895604987650026068, −5.06789549287001171042704476837, −4.04162981117684246703636268795, −3.36549717124243567635437336400, −2.70840946657743029789973900162, −1.31455133295711848474729786312, −0.69335963890996258284688142823,
1.01203244645369069385224843012, 1.95135902273090481369883436711, 2.71628785751583676268171396478, 3.87324294109917736546438075440, 4.18150504876170645988762449303, 5.32119202328114149382755477189, 5.82574915307380129244972395784, 6.65219025800913336137332189396, 7.06234951255601383864438966478, 8.094284567108082338531337124353
