L(s) = 1 | + (1.36 + 1.46i)2-s + (−0.265 + 3.99i)4-s − 1.87·7-s + (−6.19 + 5.06i)8-s + 15.1i·11-s + 18.1i·13-s + (−2.55 − 2.73i)14-s + (−15.8 − 2.11i)16-s − 12.6i·17-s − 28.1i·19-s + (−22.0 + 20.6i)22-s − 10.9·23-s + (−26.4 + 24.7i)26-s + (0.496 − 7.46i)28-s + 9.95·29-s + ⋯ |
L(s) = 1 | + (0.683 + 0.730i)2-s + (−0.0663 + 0.997i)4-s − 0.267·7-s + (−0.773 + 0.633i)8-s + 1.37i·11-s + 1.39i·13-s + (−0.182 − 0.195i)14-s + (−0.991 − 0.132i)16-s − 0.746i·17-s − 1.48i·19-s + (−1.00 + 0.938i)22-s − 0.475·23-s + (−1.01 + 0.952i)26-s + (0.0177 − 0.266i)28-s + 0.343·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.387319659\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.387319659\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 - 1.46i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.87T + 49T^{2} \) |
| 11 | \( 1 - 15.1iT - 121T^{2} \) |
| 13 | \( 1 - 18.1iT - 169T^{2} \) |
| 17 | \( 1 + 12.6iT - 289T^{2} \) |
| 19 | \( 1 + 28.1iT - 361T^{2} \) |
| 23 | \( 1 + 10.9T + 529T^{2} \) |
| 29 | \( 1 - 9.95T + 841T^{2} \) |
| 31 | \( 1 - 24.4iT - 961T^{2} \) |
| 37 | \( 1 + 17.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 28.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 56.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 49.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 60.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 110. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 131.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 52.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 62.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 103. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 57.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 145.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 66.7iT - 9.40e3T^{2} \) |
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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
−10.28166605167521788606205430598, −9.295408450119171709609926494121, −8.765347708832016340684283535729, −7.43994313481613363196027165287, −6.97923630479785488692884365201, −6.23927420302976890405137479635, −4.82141975657345788833784771953, −4.55362649885218613683739896824, −3.20126861394263485003549420147, −2.03993564717152787707993895898,
0.33680399542935435997067257521, 1.70976757323669288963660894515, 3.20779618829603318918688612895, 3.60986633502994425759282073062, 5.00631359213520066704322546055, 5.89462234192634146673180347535, 6.38779625695173232587433057507, 8.000840486719838970042663519650, 8.524848296593897432570235687419, 9.948276844759618269681956209683