| L(s) = 1 | + 0.514·3-s − 2.27·5-s + 7-s − 2.73·9-s − 11-s + 13-s − 1.16·15-s + 5.22·17-s − 3.46·19-s + 0.514·21-s + 6.62·23-s + 0.168·25-s − 2.95·27-s + 1.19·29-s + 6.39·31-s − 0.514·33-s − 2.27·35-s − 3.23·37-s + 0.514·39-s − 9.16·41-s − 4.90·43-s + 6.21·45-s − 7.40·47-s + 49-s + 2.68·51-s + 4.70·53-s + 2.27·55-s + ⋯ |
| L(s) = 1 | + 0.296·3-s − 1.01·5-s + 0.377·7-s − 0.911·9-s − 0.301·11-s + 0.277·13-s − 0.301·15-s + 1.26·17-s − 0.794·19-s + 0.112·21-s + 1.38·23-s + 0.0337·25-s − 0.567·27-s + 0.222·29-s + 1.14·31-s − 0.0895·33-s − 0.384·35-s − 0.531·37-s + 0.0823·39-s − 1.43·41-s − 0.748·43-s + 0.927·45-s − 1.08·47-s + 0.142·49-s + 0.376·51-s + 0.646·53-s + 0.306·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.468340683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468340683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.514T + 3T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 6.62T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 - 6.39T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 + 7.40T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 - 5.05T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 - 4.52T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 2.01T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 + 3.59T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 4.52T + 97T^{2} \) |
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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
−7.985928215857530865776961739123, −7.26251625921232574158938772231, −6.56362142538403553655548672050, −5.64128186700258554099014123343, −5.02830886657499728028603201391, −4.25252802383414238016083063312, −3.33069029629725196003224943449, −2.96841977081604499181709068420, −1.76716985775636915796729828138, −0.58695243049647890888319026012,
0.58695243049647890888319026012, 1.76716985775636915796729828138, 2.96841977081604499181709068420, 3.33069029629725196003224943449, 4.25252802383414238016083063312, 5.02830886657499728028603201391, 5.64128186700258554099014123343, 6.56362142538403553655548672050, 7.26251625921232574158938772231, 7.985928215857530865776961739123
