| L(s) = 1 | + 2-s + 2.86·3-s + 4-s + 5-s + 2.86·6-s − 7-s + 8-s + 5.18·9-s + 10-s + 1.53·11-s + 2.86·12-s − 4.64·13-s − 14-s + 2.86·15-s + 16-s + 17-s + 5.18·18-s + 3.38·19-s + 20-s − 2.86·21-s + 1.53·22-s − 4.58·23-s + 2.86·24-s + 25-s − 4.64·26-s + 6.24·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.65·3-s + 0.5·4-s + 0.447·5-s + 1.16·6-s − 0.377·7-s + 0.353·8-s + 1.72·9-s + 0.316·10-s + 0.463·11-s + 0.825·12-s − 1.28·13-s − 0.267·14-s + 0.738·15-s + 0.250·16-s + 0.242·17-s + 1.22·18-s + 0.777·19-s + 0.223·20-s − 0.624·21-s + 0.327·22-s − 0.955·23-s + 0.583·24-s + 0.200·25-s − 0.911·26-s + 1.20·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.504148054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.504148054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.86T + 3T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 19 | \( 1 - 3.38T + 19T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 - 1.78T + 29T^{2} \) |
| 31 | \( 1 - 0.612T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 + 8.71T + 47T^{2} \) |
| 53 | \( 1 + 4.98T + 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 7.59T + 73T^{2} \) |
| 79 | \( 1 + 8.79T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 2.79T + 89T^{2} \) |
| 97 | \( 1 + 7.85T + 97T^{2} \) |
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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
−9.753630731049717928385434137690, −9.017332862653478977518842039550, −8.042830521347081075292093701969, −7.34978051321325346253734771370, −6.54819055971674431781373682398, −5.38843885019116378034504966746, −4.35910273410778379261240485843, −3.41030684088978459581225055738, −2.67529598283467373257085496047, −1.73817018236209421487310203037,
1.73817018236209421487310203037, 2.67529598283467373257085496047, 3.41030684088978459581225055738, 4.35910273410778379261240485843, 5.38843885019116378034504966746, 6.54819055971674431781373682398, 7.34978051321325346253734771370, 8.042830521347081075292093701969, 9.017332862653478977518842039550, 9.753630731049717928385434137690
