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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6377875993\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6377875993\) |
\(L(2)\) |
|
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
−9.359017333053548233964419616108, −8.939796255963966489008711846192, −8.159512665314959886606611233463, −7.02910370164558212393915952148, −6.34488333763051150939992686890, −4.91418904143473722236416813843, −3.81137790759891345505094155881, −2.93181184986356759211701750518, −1.67619753287759528951530976142, −0.28107617985868158022666400598,
1.26931532502127418199664839518, 2.66822065978247406173551425153, 4.21326056364641063117313023995, 5.28868534391275945203639968305, 5.99834894021096091256220401521, 7.13033670670638872499019007888, 7.65132031698095782923167611577, 8.490116012510448214012379302181, 9.565666439565428760924044042929, 10.01637207208898929883112994928