| L(s) = 1 | − 2.80·3-s + 0.844·5-s + 29.0·7-s − 19.1·9-s − 17.1·11-s + 68.6·13-s − 2.36·15-s − 86.7·17-s − 77.5·19-s − 81.5·21-s − 70.2·23-s − 124.·25-s + 129.·27-s + 89.6·29-s + 8.86·31-s + 48.1·33-s + 24.5·35-s + 30.7·37-s − 192.·39-s − 153.·41-s + 171.·43-s − 16.1·45-s − 99.9·47-s + 502.·49-s + 243.·51-s + 550.·53-s − 14.4·55-s + ⋯ |
| L(s) = 1 | − 0.539·3-s + 0.0754·5-s + 1.57·7-s − 0.708·9-s − 0.470·11-s + 1.46·13-s − 0.0407·15-s − 1.23·17-s − 0.936·19-s − 0.847·21-s − 0.636·23-s − 0.994·25-s + 0.922·27-s + 0.574·29-s + 0.0513·31-s + 0.253·33-s + 0.118·35-s + 0.136·37-s − 0.791·39-s − 0.583·41-s + 0.606·43-s − 0.0534·45-s − 0.310·47-s + 1.46·49-s + 0.667·51-s + 1.42·53-s − 0.0354·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2.80T + 27T^{2} \) |
| 5 | \( 1 - 0.844T + 125T^{2} \) |
| 7 | \( 1 - 29.0T + 343T^{2} \) |
| 11 | \( 1 + 17.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 86.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 70.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 89.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 8.86T + 2.97e4T^{2} \) |
| 37 | \( 1 - 30.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 153.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 99.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 550.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 459.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 0.479T + 2.26e5T^{2} \) |
| 67 | \( 1 + 799.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 419.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 374.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 705.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 4.72T + 7.04e5T^{2} \) |
| 97 | \( 1 - 379.T + 9.12e5T^{2} \) |
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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
−8.724191104025912122153648523765, −8.533496143838043864761223408388, −7.60770799479495672990810484102, −6.34782450285838705659137841678, −5.76965384033654995545992291863, −4.78589563538921288811627330014, −4.01491007875841917438435227303, −2.46314318190157438604371392715, −1.44546828795848968364978880147, 0,
1.44546828795848968364978880147, 2.46314318190157438604371392715, 4.01491007875841917438435227303, 4.78589563538921288811627330014, 5.76965384033654995545992291863, 6.34782450285838705659137841678, 7.60770799479495672990810484102, 8.533496143838043864761223408388, 8.724191104025912122153648523765
