| L(s) = 1 | + 2·2-s − 5.87·3-s + 4·4-s + 5.19·5-s − 11.7·6-s + 8·8-s + 7.46·9-s + 10.3·10-s + 68.4·11-s − 23.4·12-s + 36.2·13-s − 30.5·15-s + 16·16-s + 95.5·17-s + 14.9·18-s + 155.·19-s + 20.7·20-s + 136.·22-s − 23·23-s − 46.9·24-s − 97.9·25-s + 72.5·26-s + 114.·27-s + 91.0·29-s − 61.0·30-s − 4.39·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.12·3-s + 0.5·4-s + 0.464·5-s − 0.798·6-s + 0.353·8-s + 0.276·9-s + 0.328·10-s + 1.87·11-s − 0.564·12-s + 0.773·13-s − 0.525·15-s + 0.250·16-s + 1.36·17-s + 0.195·18-s + 1.87·19-s + 0.232·20-s + 1.32·22-s − 0.208·23-s − 0.399·24-s − 0.783·25-s + 0.547·26-s + 0.817·27-s + 0.583·29-s − 0.371·30-s − 0.0254·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.673252836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.673252836\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 5.87T + 27T^{2} \) |
| 5 | \( 1 - 5.19T + 125T^{2} \) |
| 11 | \( 1 - 68.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 91.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4.39T + 2.97e4T^{2} \) |
| 37 | \( 1 + 35.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 358.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 31.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 273.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 331.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 404.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 420.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 559.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 499.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 674.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 589.T + 9.12e5T^{2} \) |
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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
−8.735378180852771381167810293167, −7.64804061913495451561863092394, −6.78677330343050218066805499112, −6.06089465061674383731285698741, −5.69556746121460533052805777666, −4.86252360041885227907729318577, −3.82822456509233390444437777545, −3.13305241325177579404847802736, −1.51290154605607732365178241570, −0.937331643827834284379011019672,
0.937331643827834284379011019672, 1.51290154605607732365178241570, 3.13305241325177579404847802736, 3.82822456509233390444437777545, 4.86252360041885227907729318577, 5.69556746121460533052805777666, 6.06089465061674383731285698741, 6.78677330343050218066805499112, 7.64804061913495451561863092394, 8.735378180852771381167810293167
