| L(s) = 1 | + 0.291·3-s − 0.189·5-s − 2.91·9-s − 1.52·11-s + 13-s − 0.0552·15-s − 4.92·17-s − 4.88·19-s − 1.31·23-s − 4.96·25-s − 1.72·27-s + 0.786·29-s + 5.49·31-s − 0.443·33-s + 6.89·37-s + 0.291·39-s + 4.38·41-s + 9.19·43-s + 0.553·45-s + 4.09·47-s − 1.43·51-s + 3.15·53-s + 0.289·55-s − 1.42·57-s + 2.97·59-s + 5.17·61-s − 0.189·65-s + ⋯ |
| L(s) = 1 | + 0.168·3-s − 0.0849·5-s − 0.971·9-s − 0.459·11-s + 0.277·13-s − 0.0142·15-s − 1.19·17-s − 1.12·19-s − 0.273·23-s − 0.992·25-s − 0.331·27-s + 0.145·29-s + 0.987·31-s − 0.0772·33-s + 1.13·37-s + 0.0466·39-s + 0.684·41-s + 1.40·43-s + 0.0825·45-s + 0.597·47-s − 0.200·51-s + 0.432·53-s + 0.0390·55-s − 0.188·57-s + 0.387·59-s + 0.663·61-s − 0.0235·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.344613800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.344613800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.291T + 3T^{2} \) |
| 5 | \( 1 + 0.189T + 5T^{2} \) |
| 11 | \( 1 + 1.52T + 11T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 19 | \( 1 + 4.88T + 19T^{2} \) |
| 23 | \( 1 + 1.31T + 23T^{2} \) |
| 29 | \( 1 - 0.786T + 29T^{2} \) |
| 31 | \( 1 - 5.49T + 31T^{2} \) |
| 37 | \( 1 - 6.89T + 37T^{2} \) |
| 41 | \( 1 - 4.38T + 41T^{2} \) |
| 43 | \( 1 - 9.19T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 - 3.15T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 + 8.26T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 2.28T + 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 2.92T + 97T^{2} \) |
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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
−8.135069871035093993217542971808, −7.77443401637293871579827821309, −6.62465725013541058527567657077, −6.13928258649426936680031113634, −5.39056710248977559727829286772, −4.40525208857361788937182943001, −3.83605002863325017629919349854, −2.61566894837411111085849955376, −2.21462061203923736045397477494, −0.59387237780479154110289896861,
0.59387237780479154110289896861, 2.21462061203923736045397477494, 2.61566894837411111085849955376, 3.83605002863325017629919349854, 4.40525208857361788937182943001, 5.39056710248977559727829286772, 6.13928258649426936680031113634, 6.62465725013541058527567657077, 7.77443401637293871579827821309, 8.135069871035093993217542971808
