| L(s) = 1 | + 2.79·3-s + 2·5-s + 0.208·7-s + 4.79·9-s − 11-s − 13-s + 5.58·15-s − 4·17-s + 1.20·19-s + 0.582·21-s + 0.208·23-s − 25-s + 4.99·27-s − 3.58·29-s − 2.79·33-s + 0.417·35-s + 4·37-s − 2.79·39-s + 1.79·41-s − 7.16·43-s + 9.58·45-s + 3.58·47-s − 6.95·49-s − 11.1·51-s + 4.37·53-s − 2·55-s + 3.37·57-s + ⋯ |
| L(s) = 1 | + 1.61·3-s + 0.894·5-s + 0.0788·7-s + 1.59·9-s − 0.301·11-s − 0.277·13-s + 1.44·15-s − 0.970·17-s + 0.277·19-s + 0.127·21-s + 0.0435·23-s − 0.200·25-s + 0.962·27-s − 0.665·29-s − 0.485·33-s + 0.0705·35-s + 0.657·37-s − 0.446·39-s + 0.279·41-s − 1.09·43-s + 1.42·45-s + 0.522·47-s − 0.993·49-s − 1.56·51-s + 0.600·53-s − 0.269·55-s + 0.446·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.703776568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.703776568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 0.208T + 7T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 - 0.208T + 23T^{2} \) |
| 29 | \( 1 + 3.58T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 + 7.16T + 43T^{2} \) |
| 47 | \( 1 - 3.58T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 5.58T + 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 0.626T + 83T^{2} \) |
| 89 | \( 1 - 9.58T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−10.42103804389571162993561486325, −9.602018480438196260403816148491, −9.080700111079183748901799241838, −8.178804428858095212699489940243, −7.38431499598263411043678715046, −6.30112485979134126041882408394, −5.04000200800760959848425364888, −3.82828295916896210133280689034, −2.66426558161040768959984593821, −1.86178146462217095853342049132,
1.86178146462217095853342049132, 2.66426558161040768959984593821, 3.82828295916896210133280689034, 5.04000200800760959848425364888, 6.30112485979134126041882408394, 7.38431499598263411043678715046, 8.178804428858095212699489940243, 9.080700111079183748901799241838, 9.602018480438196260403816148491, 10.42103804389571162993561486325
