| L(s) = 1 | + 2.18·3-s − 2.05·5-s + 7-s + 1.78·9-s − 11-s − 13-s − 4.50·15-s + 1.94·17-s + 2.63·19-s + 2.18·21-s − 2.37·23-s − 0.757·25-s − 2.65·27-s + 8.17·29-s + 0.915·31-s − 2.18·33-s − 2.05·35-s + 0.853·37-s − 2.18·39-s + 6.33·41-s − 8.24·43-s − 3.67·45-s − 4.20·47-s + 49-s + 4.24·51-s + 6.84·53-s + 2.05·55-s + ⋯ |
| L(s) = 1 | + 1.26·3-s − 0.921·5-s + 0.377·7-s + 0.595·9-s − 0.301·11-s − 0.277·13-s − 1.16·15-s + 0.470·17-s + 0.604·19-s + 0.477·21-s − 0.495·23-s − 0.151·25-s − 0.511·27-s + 1.51·29-s + 0.164·31-s − 0.380·33-s − 0.348·35-s + 0.140·37-s − 0.350·39-s + 0.989·41-s − 1.25·43-s − 0.548·45-s − 0.613·47-s + 0.142·49-s + 0.594·51-s + 0.940·53-s + 0.277·55-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
| \(L(1)\) |
\(\approx\) |
\(2.671550881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.671550881\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 - 2.63T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 - 0.915T + 31T^{2} \) |
| 37 | \( 1 - 0.853T + 37T^{2} \) |
| 41 | \( 1 - 6.33T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 + 4.20T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 2.58T + 71T^{2} \) |
| 73 | \( 1 + 9.32T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 3.62T + 83T^{2} \) |
| 89 | \( 1 + 7.74T + 89T^{2} \) |
| 97 | \( 1 - 4.78T + 97T^{2} \) |
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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
−7.974261582306875417037006588466, −7.43609375514746182365880967305, −6.70046672858981551277919136008, −5.66099549780524423442750464620, −4.88243675539523137050735881551, −4.08344876876554845633130017447, −3.46971637601482508249585324428, −2.76450242933116034936924820803, −1.99599700706837008135856648721, −0.75236697924852741213140753057,
0.75236697924852741213140753057, 1.99599700706837008135856648721, 2.76450242933116034936924820803, 3.46971637601482508249585324428, 4.08344876876554845633130017447, 4.88243675539523137050735881551, 5.66099549780524423442750464620, 6.70046672858981551277919136008, 7.43609375514746182365880967305, 7.974261582306875417037006588466
