| L(s) = 1 | + 1.46·2-s − 0.114i·3-s − 1.86·4-s − 0.167i·6-s + 4.56·7-s − 8.56·8-s + 8.98·9-s + (5.89 + 9.28i)11-s + 0.213i·12-s + 16.5·13-s + 6.66·14-s − 5.05·16-s + 17.2·17-s + 13.1·18-s − 35.8i·19-s + ⋯ |
| L(s) = 1 | + 0.730·2-s − 0.0381i·3-s − 0.466·4-s − 0.0278i·6-s + 0.651·7-s − 1.07·8-s + 0.998·9-s + (0.535 + 0.844i)11-s + 0.0178i·12-s + 1.27·13-s + 0.475·14-s − 0.316·16-s + 1.01·17-s + 0.729·18-s − 1.88i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.31032 + 0.117532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31032 + 0.117532i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (-5.89 - 9.28i)T \) |
| good | 2 | \( 1 - 1.46T + 4T^{2} \) |
| 3 | \( 1 + 0.114iT - 9T^{2} \) |
| 7 | \( 1 - 4.56T + 49T^{2} \) |
| 13 | \( 1 - 16.5T + 169T^{2} \) |
| 17 | \( 1 - 17.2T + 289T^{2} \) |
| 19 | \( 1 + 35.8iT - 361T^{2} \) |
| 23 | \( 1 - 29.3iT - 529T^{2} \) |
| 29 | \( 1 - 8.51iT - 841T^{2} \) |
| 31 | \( 1 + 26.3T + 961T^{2} \) |
| 37 | \( 1 - 44.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 52.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 6.77T + 1.84e3T^{2} \) |
| 47 | \( 1 - 15.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 33.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 51.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 23.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 113. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 8.00T + 5.04e3T^{2} \) |
| 73 | \( 1 + 32.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 52.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 43.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 73.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 22.0iT - 9.40e3T^{2} \) |
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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
−11.86974937433611689885994942057, −10.92353460995284708973252774006, −9.649336371058148041908358060519, −8.957407149094602211158962493935, −7.68993559651587418064669264697, −6.62967130729780375309908008588, −5.31822867061201064963382891840, −4.46081685055749145834890009067, −3.44148310287432756045825762980, −1.40598216907358845407730987868,
1.30188053156297351002580844148, 3.52555598573029618760605911792, 4.21085281389185579801346805894, 5.53554519592089515758596928598, 6.32630064794284180469930293293, 7.917588965911605649634866845433, 8.666518405723340612161967664241, 9.805883893187187918130743410827, 10.79055424313430144626929132763, 11.88093100914102275338949140894
