| L(s) = 1 | + 1.89·3-s + 3.07·5-s + 7-s + 0.603·9-s − 11-s + 13-s + 5.83·15-s + 5.38·17-s − 4.43·19-s + 1.89·21-s + 1.61·23-s + 4.44·25-s − 4.54·27-s + 7.65·29-s + 1.08·31-s − 1.89·33-s + 3.07·35-s + 2.13·37-s + 1.89·39-s + 4.92·41-s − 11.4·43-s + 1.85·45-s + 9.41·47-s + 49-s + 10.2·51-s − 0.843·53-s − 3.07·55-s + ⋯ |
| L(s) = 1 | + 1.09·3-s + 1.37·5-s + 0.377·7-s + 0.201·9-s − 0.301·11-s + 0.277·13-s + 1.50·15-s + 1.30·17-s − 1.01·19-s + 0.414·21-s + 0.336·23-s + 0.888·25-s − 0.875·27-s + 1.42·29-s + 0.194·31-s − 0.330·33-s + 0.519·35-s + 0.350·37-s + 0.303·39-s + 0.769·41-s − 1.74·43-s + 0.276·45-s + 1.37·47-s + 0.142·49-s + 1.43·51-s − 0.115·53-s − 0.414·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.984768056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.984768056\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.89T + 3T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 17 | \( 1 - 5.38T + 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 - 1.08T + 31T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 9.41T + 47T^{2} \) |
| 53 | \( 1 + 0.843T + 53T^{2} \) |
| 59 | \( 1 + 3.51T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 8.12T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 3.67T + 73T^{2} \) |
| 79 | \( 1 + 9.72T + 79T^{2} \) |
| 83 | \( 1 - 8.38T + 83T^{2} \) |
| 89 | \( 1 + 7.52T + 89T^{2} \) |
| 97 | \( 1 + 0.326T + 97T^{2} \) |
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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
−8.419084446609053416036062575311, −8.009038979938229989304087277656, −7.02303390675518945608704432467, −6.15825558984944012816430495537, −5.55230540096152986100122074685, −4.71983998968361249639756528497, −3.64574380072387456324671500674, −2.75021634315379066655213619419, −2.16807812457011578594282158652, −1.18066889434452593838810602032,
1.18066889434452593838810602032, 2.16807812457011578594282158652, 2.75021634315379066655213619419, 3.64574380072387456324671500674, 4.71983998968361249639756528497, 5.55230540096152986100122074685, 6.15825558984944012816430495537, 7.02303390675518945608704432467, 8.009038979938229989304087277656, 8.419084446609053416036062575311
