| L(s) = 1 | + 2i·3-s + (3 + 3i)7-s − 9-s + (2 − 2i)11-s + (2 + 2i)13-s − 6·17-s + (−4 − 4i)19-s + (−6 + 6i)21-s + (−5 + 5i)23-s + 4i·27-s + (−2 − 5i)29-s + (−6 + 6i)31-s + (4 + 4i)33-s − 6i·37-s + (−4 + 4i)39-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s + (1.13 + 1.13i)7-s − 0.333·9-s + (0.603 − 0.603i)11-s + (0.554 + 0.554i)13-s − 1.45·17-s + (−0.917 − 0.917i)19-s + (−1.30 + 1.30i)21-s + (−1.04 + 1.04i)23-s + 0.769i·27-s + (−0.371 − 0.928i)29-s + (−1.07 + 1.07i)31-s + (0.696 + 0.696i)33-s − 0.986i·37-s + (−0.640 + 0.640i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680634874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680634874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + (2 + 5i)T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2 + 2i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2 - 2i)T + 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (4 + 4i)T + 19iT^{2} \) |
| 23 | \( 1 + (5 - 5i)T - 23iT^{2} \) |
| 31 | \( 1 + (6 - 6i)T - 31iT^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + (-7 - 7i)T + 41iT^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + (4 - 4i)T - 53iT^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + (-9 + 9i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3 + 3i)T - 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + (8 + 8i)T + 79iT^{2} \) |
| 83 | \( 1 + (3 - 3i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7 - 7i)T + 89iT^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.193460279541839303166938199021, −8.598257970033700813560861487349, −7.83664636827551653491430050791, −6.64718889810883734104470892343, −5.92135826974054948247144656869, −5.11631642745775713547612054783, −4.34089993243800108560338829412, −3.84623250571893868903877262199, −2.51736455920833975892081395198, −1.63097541710341661356499907220,
0.51166811453268143169466264659, 1.71591908495142271060119777248, 2.11926834235353347355197917140, 3.98061822708543514973983380934, 4.22090159364775511284628984565, 5.43093242307119684525849543106, 6.49205445146289496555708090026, 6.88636332311333885737960289917, 7.70305412137973007577212089918, 8.207258720707094290427647002058
