| 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 | \( 1 + (-3.23 - 0.464i)T + (93.0 + 27.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.39789602552580896507111476025, −10.11300788118149725675463254379, −8.838644368021332831468006189481, −7.31056917988644702597943307035, −6.97726668829172168541223571845, −5.90043543914742130181195263211, −4.46800918512330313882453346156, −3.56962143864445421417141710008, −2.66252579710544793445756014271, −0.988529803559772992182338768422,
2.52973978602482203022282603969, 3.34046597543296257448581327888, 4.73828724308473674711361769052, 5.40846565293325222638660251330, 6.33749701276641353456871085495, 7.48805353005394330264358978431, 8.514318250913816189415047416985, 9.122999923937559254625708763738, 9.933456222028095141686721491364, 11.56007275795069746664940219281
