| L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.978 + 0.207i)3-s + (−0.809 + 0.587i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−4.59 − 2.04i)7-s + (0.809 + 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.978 − 0.207i)10-s + (0.443 + 4.21i)11-s + (0.669 − 0.743i)12-s + (4.36 + 4.84i)13-s + (−0.525 + 5.00i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.108 + 1.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.564 + 0.120i)3-s + (−0.404 + 0.293i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (−1.73 − 0.773i)7-s + (0.286 + 0.207i)8-s + (0.304 − 0.135i)9-s + (−0.309 − 0.0657i)10-s + (0.133 + 1.27i)11-s + (0.193 − 0.214i)12-s + (1.21 + 1.34i)13-s + (−0.140 + 1.33i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (−0.0261 + 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.870611 - 0.295235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.870611 - 0.295235i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.978 - 0.207i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-4.41 - 3.39i)T \) |
| good | 7 | \( 1 + (4.59 + 2.04i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.443 - 4.21i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-4.36 - 4.84i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.108 - 1.02i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-2.38 + 2.64i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (2.20 + 1.60i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.89 + 5.82i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (3.78 + 6.54i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.7 - 2.28i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-1.47 + 1.64i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-1.97 + 6.08i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-10.7 + 4.79i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-7.24 + 1.53i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + (4.94 - 8.56i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.54 - 2.46i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.872 - 8.29i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (0.869 - 8.27i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-5.43 - 1.15i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-2.70 + 1.96i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (6.87 - 4.99i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.906164905158288945152892298781, −9.494210464279497385990452399867, −8.667562188044381124383579715925, −7.16943357048371555656828095086, −6.67844870101216103134160324336, −5.67700548390423346816484450670, −4.19974518260252650574430134982, −3.90836225874244325207952741626, −2.30958646552706323320643711572, −0.845152193779645718138389368640,
0.77198760396734362926990723082, 2.95227294073865865213367087648, 3.66494842726434622146331034666, 5.55641561331599480922466701091, 5.91373557412020438479058480228, 6.43008057588184529369047883406, 7.52621696067243023321549959533, 8.523314780985643132249627114788, 9.257613476372454520499270367153, 10.13326576343314213581002813563
