L(s) = 1 | − 2.23i·5-s − 0.246i·7-s + 3.93i·11-s − 16.2·13-s − 9.26i·17-s + 9.27i·19-s + (−11.8 + 19.6i)23-s − 5.00·25-s − 12.7·29-s − 18.4·31-s − 0.551·35-s + 32.9i·37-s + 16.7·41-s − 71.8i·43-s + 72.2·47-s + ⋯ |
L(s) = 1 | − 0.447i·5-s − 0.0352i·7-s + 0.357i·11-s − 1.25·13-s − 0.545i·17-s + 0.488i·19-s + (−0.516 + 0.856i)23-s − 0.200·25-s − 0.438·29-s − 0.595·31-s − 0.0157·35-s + 0.889i·37-s + 0.408·41-s − 1.67i·43-s + 1.53·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.540606030\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.540606030\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (11.8 - 19.6i)T \) |
good | 7 | \( 1 + 0.246iT - 49T^{2} \) |
| 11 | \( 1 - 3.93iT - 121T^{2} \) |
| 13 | \( 1 + 16.2T + 169T^{2} \) |
| 17 | \( 1 + 9.26iT - 289T^{2} \) |
| 19 | \( 1 - 9.27iT - 361T^{2} \) |
| 29 | \( 1 + 12.7T + 841T^{2} \) |
| 31 | \( 1 + 18.4T + 961T^{2} \) |
| 37 | \( 1 - 32.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 16.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 71.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 72.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 8.57iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 68.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 5.62iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 72.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 100.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 115.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.109iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 27.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 59.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 160. iT - 9.40e3T^{2} \) |
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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
−8.110342460224549541407168030340, −7.40316321849396038028821761302, −6.95373139619130676176331533377, −5.75308755904237115928004933181, −5.28587330224226485540745169229, −4.42061609416387250652084224278, −3.67142439922806982822260410653, −2.54945084924300021365541882928, −1.74779400465273788326085375835, −0.48857385508696466773140570209,
0.62698481230745419440563274639, 2.07289095087139622505879853856, 2.70603529337467514659273692625, 3.73355372212944740589362252736, 4.51109878602846456765229605506, 5.38816837251840035747552716015, 6.14237915471731270144924888501, 6.86608055150756151827486006280, 7.61673608383709294785533252043, 8.152132292880156968210705077258