| L(s) = 1 | + 0.220·3-s + 1.20·5-s − 3.77·7-s − 2.95·9-s − 3.28·11-s − 2.57·13-s + 0.265·15-s + 2.82·17-s + 0.306·19-s − 0.830·21-s + 8.21·23-s − 3.54·25-s − 1.31·27-s + 1.89·29-s − 2.28·31-s − 0.723·33-s − 4.55·35-s + 6.00·37-s − 0.565·39-s + 1.51·41-s − 1.64·43-s − 3.56·45-s − 6.84·47-s + 7.23·49-s + 0.623·51-s − 5.95·53-s − 3.96·55-s + ⋯ |
| L(s) = 1 | + 0.127·3-s + 0.539·5-s − 1.42·7-s − 0.983·9-s − 0.991·11-s − 0.712·13-s + 0.0685·15-s + 0.686·17-s + 0.0704·19-s − 0.181·21-s + 1.71·23-s − 0.708·25-s − 0.252·27-s + 0.351·29-s − 0.411·31-s − 0.126·33-s − 0.769·35-s + 0.986·37-s − 0.0906·39-s + 0.236·41-s − 0.251·43-s − 0.530·45-s − 0.998·47-s + 1.03·49-s + 0.0872·51-s − 0.818·53-s − 0.534·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.201583511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.201583511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 0.220T + 3T^{2} \) |
| 5 | \( 1 - 1.20T + 5T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 + 2.57T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.306T + 19T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 - 6.00T + 37T^{2} \) |
| 41 | \( 1 - 1.51T + 41T^{2} \) |
| 43 | \( 1 + 1.64T + 43T^{2} \) |
| 47 | \( 1 + 6.84T + 47T^{2} \) |
| 53 | \( 1 + 5.95T + 53T^{2} \) |
| 59 | \( 1 - 8.38T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 4.93T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 3.13T + 79T^{2} \) |
| 83 | \( 1 - 5.57T + 83T^{2} \) |
| 89 | \( 1 - 0.0590T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.461694034559450877142989027353, −7.71713709866718914266330153902, −6.92821580058006617960005738404, −6.19199416634074795931345529229, −5.50419709373123703432492361776, −4.91663033356005445402678618770, −3.52373842325663698870740926531, −2.93955601899222576290494956768, −2.26909155564019116363504077768, −0.58655067339709359803186272060,
0.58655067339709359803186272060, 2.26909155564019116363504077768, 2.93955601899222576290494956768, 3.52373842325663698870740926531, 4.91663033356005445402678618770, 5.50419709373123703432492361776, 6.19199416634074795931345529229, 6.92821580058006617960005738404, 7.71713709866718914266330153902, 8.461694034559450877142989027353
