| L(s) = 1 | + 34·7-s − 16·11-s − 58·13-s − 70·17-s + 4·19-s − 134·23-s + 242·29-s + 100·31-s + 438·37-s + 138·41-s − 178·43-s + 22·47-s + 813·49-s + 162·53-s + 268·59-s + 250·61-s − 422·67-s + 852·71-s − 306·73-s − 544·77-s − 456·79-s + 434·83-s + 726·89-s − 1.97e3·91-s − 1.37e3·97-s − 126·101-s + 1.26e3·103-s + ⋯ |
| L(s) = 1 | + 1.83·7-s − 0.438·11-s − 1.23·13-s − 0.998·17-s + 0.0482·19-s − 1.21·23-s + 1.54·29-s + 0.579·31-s + 1.94·37-s + 0.525·41-s − 0.631·43-s + 0.0682·47-s + 2.37·49-s + 0.419·53-s + 0.591·59-s + 0.524·61-s − 0.769·67-s + 1.42·71-s − 0.490·73-s − 0.805·77-s − 0.649·79-s + 0.573·83-s + 0.864·89-s − 2.27·91-s − 1.44·97-s − 0.124·101-s + 1.20·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.422607209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.422607209\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 134 T + p^{3} T^{2} \) |
| 29 | \( 1 - 242 T + p^{3} T^{2} \) |
| 31 | \( 1 - 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 438 T + p^{3} T^{2} \) |
| 41 | \( 1 - 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 178 T + p^{3} T^{2} \) |
| 47 | \( 1 - 22 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 268 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 422 T + p^{3} T^{2} \) |
| 71 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 73 | \( 1 + 306 T + p^{3} T^{2} \) |
| 79 | \( 1 + 456 T + p^{3} T^{2} \) |
| 83 | \( 1 - 434 T + p^{3} T^{2} \) |
| 89 | \( 1 - 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1378 T + p^{3} T^{2} \) |
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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
−8.714920937004332375989809296385, −8.062064826973896385444177167705, −7.57705332200855523271661116511, −6.57116141015434663179116700405, −5.53215421120834369451555315989, −4.67177149718325533648591936663, −4.31697964555551748513781128304, −2.62791102696623370404111527333, −2.00370736137998008116755274441, −0.73120802362207737469665229099,
0.73120802362207737469665229099, 2.00370736137998008116755274441, 2.62791102696623370404111527333, 4.31697964555551748513781128304, 4.67177149718325533648591936663, 5.53215421120834369451555315989, 6.57116141015434663179116700405, 7.57705332200855523271661116511, 8.062064826973896385444177167705, 8.714920937004332375989809296385
