L(s) = 1 | + (−0.209 − 0.209i)3-s + 1.73·7-s − 2.91i·9-s + (−0.505 − 0.505i)11-s + (−1.88 − 1.88i)13-s − 4.53i·17-s + (−3.22 + 3.22i)19-s + (−0.364 − 0.364i)21-s − 8.85·23-s + (−1.23 + 1.23i)27-s + (2.44 − 2.44i)29-s + 5.70·31-s + 0.211i·33-s + (−5.35 + 5.35i)37-s + 0.791i·39-s + ⋯ |
L(s) = 1 | + (−0.120 − 0.120i)3-s + 0.656·7-s − 0.970i·9-s + (−0.152 − 0.152i)11-s + (−0.523 − 0.523i)13-s − 1.09i·17-s + (−0.738 + 0.738i)19-s + (−0.0794 − 0.0794i)21-s − 1.84·23-s + (−0.238 + 0.238i)27-s + (0.453 − 0.453i)29-s + 1.02·31-s + 0.0368i·33-s + (−0.880 + 0.880i)37-s + 0.126i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9719322666\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9719322666\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.209 + 0.209i)T + 3iT^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + (0.505 + 0.505i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.88 + 1.88i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.53iT - 17T^{2} \) |
| 19 | \( 1 + (3.22 - 3.22i)T - 19iT^{2} \) |
| 23 | \( 1 + 8.85T + 23T^{2} \) |
| 29 | \( 1 + (-2.44 + 2.44i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 + (5.35 - 5.35i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.0iT - 41T^{2} \) |
| 43 | \( 1 + (2.10 - 2.10i)T - 43iT^{2} \) |
| 47 | \( 1 - 4.32iT - 47T^{2} \) |
| 53 | \( 1 + (-1.37 + 1.37i)T - 53iT^{2} \) |
| 59 | \( 1 + (-6.64 - 6.64i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.26 + 5.26i)T - 61iT^{2} \) |
| 67 | \( 1 + (10.5 + 10.5i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.0iT - 71T^{2} \) |
| 73 | \( 1 + 6.63T + 73T^{2} \) |
| 79 | \( 1 - 4.27T + 79T^{2} \) |
| 83 | \( 1 + (9.15 + 9.15i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.23iT - 89T^{2} \) |
| 97 | \( 1 + 1.94iT - 97T^{2} \) |
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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
−9.089439148636274357532740821778, −8.222615994904619435923713281806, −7.65089823599586917607568235170, −6.61213906833062397003505987292, −5.92190584942662086734538646151, −4.96634491244926493849610117260, −4.08194794072847895872238938287, −3.01204230293955839421973603380, −1.83366947049544045500317075421, −0.36155793031845857230814833435,
1.71358854951967799712861965243, 2.50038647412054455724135330416, 4.05311133585902043796145005803, 4.66096302407712050504327879642, 5.53133619804780834962574626676, 6.47868031082616471151074439294, 7.37502421369225768788703614934, 8.267714847205402208458767030159, 8.618681351334654689470637561779, 10.02524529153831672234056382354