L(s) = 1 | − 1.32·3-s + 2.33·5-s − 2.12·7-s − 1.24·9-s − 1.69·11-s − 4.85·13-s − 3.09·15-s + 7.73·17-s + 0.742·19-s + 2.80·21-s − 2.11·23-s + 0.475·25-s + 5.62·27-s + 4.46·29-s − 3.40·31-s + 2.24·33-s − 4.96·35-s + 0.982·37-s + 6.41·39-s + 7.22·41-s − 4.18·43-s − 2.92·45-s + 4.64·47-s − 2.49·49-s − 10.2·51-s + 10.5·53-s − 3.97·55-s + ⋯ |
L(s) = 1 | − 0.764·3-s + 1.04·5-s − 0.802·7-s − 0.416·9-s − 0.511·11-s − 1.34·13-s − 0.799·15-s + 1.87·17-s + 0.170·19-s + 0.612·21-s − 0.440·23-s + 0.0950·25-s + 1.08·27-s + 0.829·29-s − 0.610·31-s + 0.390·33-s − 0.839·35-s + 0.161·37-s + 1.02·39-s + 1.12·41-s − 0.638·43-s − 0.435·45-s + 0.677·47-s − 0.356·49-s − 1.43·51-s + 1.44·53-s − 0.535·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 1.32T + 3T^{2} \) |
| 5 | \( 1 - 2.33T + 5T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 7.73T + 17T^{2} \) |
| 19 | \( 1 - 0.742T + 19T^{2} \) |
| 23 | \( 1 + 2.11T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 - 0.982T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 - 4.64T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 0.980T + 59T^{2} \) |
| 61 | \( 1 - 2.61T + 61T^{2} \) |
| 67 | \( 1 - 9.59T + 67T^{2} \) |
| 71 | \( 1 + 9.87T + 71T^{2} \) |
| 73 | \( 1 - 4.75T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 8.39T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 8.72T + 97T^{2} \) |
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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
−7.42037484764770048260485087596, −6.62161380296341946476233117789, −5.98149174479237732604573150688, −5.40514040580830628584651282645, −5.10734519336812405731859111325, −3.88820851172672376318895644788, −2.88158949428259582996912579826, −2.39622179520850945859735760994, −1.10618681439222227272551764856, 0,
1.10618681439222227272551764856, 2.39622179520850945859735760994, 2.88158949428259582996912579826, 3.88820851172672376318895644788, 5.10734519336812405731859111325, 5.40514040580830628584651282645, 5.98149174479237732604573150688, 6.62161380296341946476233117789, 7.42037484764770048260485087596