L(s) = 1 | − 1.73i·2-s − 0.999·4-s + 2i·7-s − 1.73i·8-s + 3.46·11-s + i·13-s + 3.46·14-s − 5·16-s − 5.19i·17-s − 2·19-s − 5.99i·22-s + 3.46i·23-s + 1.73·26-s − 1.99i·28-s + 1.73·29-s + ⋯ |
L(s) = 1 | − 1.22i·2-s − 0.499·4-s + 0.755i·7-s − 0.612i·8-s + 1.04·11-s + 0.277i·13-s + 0.925·14-s − 1.25·16-s − 1.26i·17-s − 0.458·19-s − 1.27i·22-s + 0.722i·23-s + 0.339·26-s − 0.377i·28-s + 0.321·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.925394283\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.925394283\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 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
−9.257060689671249844372639390695, −8.428212527773036362402408644431, −7.22509140921732545139510650108, −6.57886520754154553217189438707, −5.62171107945741365620657903141, −4.53722064005510145192553450233, −3.74740179583538483522747141476, −2.75776967402327200214552043673, −2.01098669195593130483312995182, −0.800163558692170654301047797943,
1.17415526970609695804262739736, 2.60190979083153458403636109218, 3.99027584132772433523918590212, 4.55684446468755838420231779525, 5.69967325207386727261406353477, 6.51566896035934009893868840447, 6.76655596086778413640290626364, 7.909705775426582483682171476690, 8.289321427004824918516686522752, 9.120925364057106426532125792343