| L(s) = 1 | + 2.25·2-s + 3-s + 3.08·4-s + 3.54·5-s + 2.25·6-s + 1.54·7-s + 2.43·8-s + 9-s + 7.99·10-s − 1.30·11-s + 3.08·12-s + 13-s + 3.48·14-s + 3.54·15-s − 0.665·16-s − 0.968·17-s + 2.25·18-s + 3.57·19-s + 10.9·20-s + 1.54·21-s − 2.95·22-s − 1.40·23-s + 2.43·24-s + 7.58·25-s + 2.25·26-s + 27-s + 4.76·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.577·3-s + 1.54·4-s + 1.58·5-s + 0.920·6-s + 0.584·7-s + 0.862·8-s + 0.333·9-s + 2.52·10-s − 0.394·11-s + 0.889·12-s + 0.277·13-s + 0.932·14-s + 0.916·15-s − 0.166·16-s − 0.234·17-s + 0.531·18-s + 0.820·19-s + 2.44·20-s + 0.337·21-s − 0.629·22-s − 0.291·23-s + 0.497·24-s + 1.51·25-s + 0.442·26-s + 0.192·27-s + 0.901·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.317176605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.317176605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.25T + 2T^{2} \) |
| 5 | \( 1 - 3.54T + 5T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 + 1.30T + 11T^{2} \) |
| 17 | \( 1 + 0.968T + 17T^{2} \) |
| 19 | \( 1 - 3.57T + 19T^{2} \) |
| 23 | \( 1 + 1.40T + 23T^{2} \) |
| 29 | \( 1 + 1.54T + 29T^{2} \) |
| 31 | \( 1 + 6.81T + 31T^{2} \) |
| 37 | \( 1 - 0.115T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 + 9.39T + 47T^{2} \) |
| 53 | \( 1 + 2.40T + 53T^{2} \) |
| 59 | \( 1 + 9.66T + 59T^{2} \) |
| 61 | \( 1 + 4.09T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 4.08T + 71T^{2} \) |
| 73 | \( 1 - 8.93T + 73T^{2} \) |
| 79 | \( 1 - 3.82T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 8.61T + 97T^{2} \) |
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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
−8.368255352145851498970292185754, −7.52731261601012027509962024433, −6.66631303058579137914927747413, −5.99327976993682423548578486110, −5.34693898079499417234053512413, −4.84508998969510852908287357430, −3.85997676110671549579513921342, −3.00605669167758883408196949161, −2.23181278879099629145672859049, −1.55742459616184167998513330716,
1.55742459616184167998513330716, 2.23181278879099629145672859049, 3.00605669167758883408196949161, 3.85997676110671549579513921342, 4.84508998969510852908287357430, 5.34693898079499417234053512413, 5.99327976993682423548578486110, 6.66631303058579137914927747413, 7.52731261601012027509962024433, 8.368255352145851498970292185754
