| L(s) = 1 | + 1.67·3-s − 1.19·7-s − 0.193·9-s − 5.44·11-s − 13-s − 6.96·17-s − 0.869·19-s − 2·21-s − 0.324·23-s − 5.35·27-s − 4.15·29-s + 2.09·31-s − 9.11·33-s + 1.50·37-s − 1.67·39-s + 9.92·41-s + 7.86·43-s − 6.80·47-s − 5.57·49-s − 11.6·51-s − 11.3·53-s − 1.45·57-s + 9.18·59-s + 9.43·61-s + 0.231·63-s + 7.11·67-s − 0.544·69-s + ⋯ |
| L(s) = 1 | + 0.967·3-s − 0.451·7-s − 0.0646·9-s − 1.64·11-s − 0.277·13-s − 1.68·17-s − 0.199·19-s − 0.436·21-s − 0.0677·23-s − 1.02·27-s − 0.771·29-s + 0.375·31-s − 1.58·33-s + 0.247·37-s − 0.268·39-s + 1.54·41-s + 1.19·43-s − 0.992·47-s − 0.796·49-s − 1.63·51-s − 1.55·53-s − 0.192·57-s + 1.19·59-s + 1.20·61-s + 0.0291·63-s + 0.869·67-s − 0.0655·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 + 5.44T + 11T^{2} \) |
| 17 | \( 1 + 6.96T + 17T^{2} \) |
| 19 | \( 1 + 0.869T + 19T^{2} \) |
| 23 | \( 1 + 0.324T + 23T^{2} \) |
| 29 | \( 1 + 4.15T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 - 1.50T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 9.18T + 59T^{2} \) |
| 61 | \( 1 - 9.43T + 61T^{2} \) |
| 67 | \( 1 - 7.11T + 67T^{2} \) |
| 71 | \( 1 - 8.21T + 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 + 2.57T + 79T^{2} \) |
| 83 | \( 1 + 2.99T + 83T^{2} \) |
| 89 | \( 1 - 9.47T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−9.265688998524649751057282006601, −8.347345404098042902307770516991, −7.85040181903324680265436199966, −6.91317564432457494402207393065, −5.91581141584203964732283105472, −4.93196689229802980956309117238, −3.89840507751529520747441852494, −2.74647125229004022720987551629, −2.26210247601236549929815619997, 0,
2.26210247601236549929815619997, 2.74647125229004022720987551629, 3.89840507751529520747441852494, 4.93196689229802980956309117238, 5.91581141584203964732283105472, 6.91317564432457494402207393065, 7.85040181903324680265436199966, 8.347345404098042902307770516991, 9.265688998524649751057282006601
