| L(s) = 1 | + 2.40·3-s − 2.20·7-s + 2.78·9-s + 1.68·11-s + 6.87·13-s − 5.86·17-s + 0.108·19-s − 5.31·21-s + 4.40·23-s − 0.513·27-s − 4.25·29-s + 9.07·31-s + 4.04·33-s + 1.74·37-s + 16.5·39-s + 7.95·41-s − 8.89·43-s + 10.5·47-s − 2.12·49-s − 14.1·51-s + 8.51·53-s + 0.260·57-s + 7.67·59-s + 1.94·61-s − 6.15·63-s + 0.788·67-s + 10.5·69-s + ⋯ |
| L(s) = 1 | + 1.38·3-s − 0.834·7-s + 0.928·9-s + 0.507·11-s + 1.90·13-s − 1.42·17-s + 0.0248·19-s − 1.15·21-s + 0.918·23-s − 0.0987·27-s − 0.790·29-s + 1.62·31-s + 0.704·33-s + 0.287·37-s + 2.64·39-s + 1.24·41-s − 1.35·43-s + 1.53·47-s − 0.302·49-s − 1.97·51-s + 1.16·53-s + 0.0345·57-s + 0.999·59-s + 0.249·61-s − 0.775·63-s + 0.0963·67-s + 1.27·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.224066610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.224066610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 6.87T + 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 - 0.108T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 + 4.25T + 29T^{2} \) |
| 31 | \( 1 - 9.07T + 31T^{2} \) |
| 37 | \( 1 - 1.74T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 8.51T + 53T^{2} \) |
| 59 | \( 1 - 7.67T + 59T^{2} \) |
| 61 | \( 1 - 1.94T + 61T^{2} \) |
| 67 | \( 1 - 0.788T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 + 4.08T + 83T^{2} \) |
| 89 | \( 1 + 2.37T + 89T^{2} \) |
| 97 | \( 1 - 5.70T + 97T^{2} \) |
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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
−8.534711056562788620986556491733, −7.994161633014397889341414554312, −6.85835220760440089928627928690, −6.51657821095068523070446606849, −5.58814440738750453402582690095, −4.24018134683575526941493658189, −3.77116765835920442107650954761, −2.98690796114780750824466484933, −2.19942232436056688573320124342, −1.00066617030322062623973204524,
1.00066617030322062623973204524, 2.19942232436056688573320124342, 2.98690796114780750824466484933, 3.77116765835920442107650954761, 4.24018134683575526941493658189, 5.58814440738750453402582690095, 6.51657821095068523070446606849, 6.85835220760440089928627928690, 7.994161633014397889341414554312, 8.534711056562788620986556491733
