| L(s) = 1 | + (1.12 + 0.862i)2-s + (0.654 + 0.755i)3-s + (0.513 + 1.93i)4-s + (−0.175 − 1.22i)5-s + (0.0825 + 1.41i)6-s + (−1.83 + 4.02i)7-s + (−1.09 + 2.60i)8-s + (−0.142 + 0.989i)9-s + (0.858 − 1.52i)10-s + (0.205 − 0.701i)11-s + (−1.12 + 1.65i)12-s + (−1.90 + 0.869i)13-s + (−5.53 + 2.92i)14-s + (0.809 − 0.934i)15-s + (−3.47 + 1.98i)16-s + (2.50 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (0.792 + 0.609i)2-s + (0.378 + 0.436i)3-s + (0.256 + 0.966i)4-s + (−0.0787 − 0.547i)5-s + (0.0336 + 0.576i)6-s + (−0.694 + 1.52i)7-s + (−0.385 + 0.922i)8-s + (−0.0474 + 0.329i)9-s + (0.271 − 0.481i)10-s + (0.0620 − 0.211i)11-s + (−0.324 + 0.477i)12-s + (−0.528 + 0.241i)13-s + (−1.47 + 0.782i)14-s + (0.209 − 0.241i)15-s + (−0.868 + 0.496i)16-s + (0.606 − 0.943i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.991806 + 1.96690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991806 + 1.96690i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.12 - 0.862i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-2.21 - 4.25i)T \) |
| good | 5 | \( 1 + (0.175 + 1.22i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (1.83 - 4.02i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.205 + 0.701i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.90 - 0.869i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.50 + 3.89i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.534 - 0.831i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.81 + 4.38i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.18 - 1.02i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.690 - 4.80i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.851 - 5.91i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.62 + 4.87i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 3.20iT - 47T^{2} \) |
| 53 | \( 1 + (-4.99 + 10.9i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.58 - 7.85i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.08 + 7.02i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (2.30 + 7.86i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (2.19 + 7.48i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (11.2 - 7.24i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (6.41 + 14.0i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-8.31 - 1.19i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (11.8 - 10.2i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-3.23 + 0.464i)T + (93.0 - 27.3i)T^{2} \) |
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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
−11.56007275795069746664940219281, −9.933456222028095141686721491364, −9.122999923937559254625708763738, −8.514318250913816189415047416985, −7.48805353005394330264358978431, −6.33749701276641353456871085495, −5.40846565293325222638660251330, −4.73828724308473674711361769052, −3.34046597543296257448581327888, −2.52973978602482203022282603969,
0.988529803559772992182338768422, 2.66252579710544793445756014271, 3.56962143864445421417141710008, 4.46800918512330313882453346156, 5.90043543914742130181195263211, 6.97726668829172168541223571845, 7.31056917988644702597943307035, 8.838644368021332831468006189481, 10.11300788118149725675463254379, 10.39789602552580896507111476025
