L(s) = 1 | + (−0.504 + 1.32i)2-s + (0.589 − 2.20i)3-s + (−1.49 − 1.33i)4-s + (−0.622 − 2.32i)5-s + (2.60 + 1.88i)6-s + (−2.41 − 1.09i)7-s + (2.51 − 1.29i)8-s + (−1.89 − 1.09i)9-s + (3.38 + 0.348i)10-s + (0.0284 + 0.00762i)11-s + (−3.81 + 2.49i)12-s + (4.38 + 4.38i)13-s + (2.65 − 2.63i)14-s − 5.47·15-s + (0.450 + 3.97i)16-s + (1.36 + 2.35i)17-s + ⋯ |
L(s) = 1 | + (−0.356 + 0.934i)2-s + (0.340 − 1.27i)3-s + (−0.745 − 0.666i)4-s + (−0.278 − 1.03i)5-s + (1.06 + 0.770i)6-s + (−0.911 − 0.412i)7-s + (0.888 − 0.459i)8-s + (−0.631 − 0.364i)9-s + (1.06 + 0.110i)10-s + (0.00857 + 0.00229i)11-s + (−1.09 + 0.720i)12-s + (1.21 + 1.21i)13-s + (0.709 − 0.704i)14-s − 1.41·15-s + (0.112 + 0.993i)16-s + (0.330 + 0.571i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.750173 - 0.363764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.750173 - 0.363764i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.504 - 1.32i)T \) |
| 7 | \( 1 + (2.41 + 1.09i)T \) |
good | 3 | \( 1 + (-0.589 + 2.20i)T + (-2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (0.622 + 2.32i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.0284 - 0.00762i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-4.38 - 4.38i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.36 - 2.35i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.73 + 1.53i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (3.33 + 1.92i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.93 + 4.93i)T + 29iT^{2} \) |
| 31 | \( 1 + (1.29 + 2.23i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.09 - 7.83i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 0.207iT - 41T^{2} \) |
| 43 | \( 1 + (0.278 - 0.278i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.91 - 3.31i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.22 - 0.328i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.208 + 0.0558i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-5.93 + 1.59i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (3.20 - 11.9i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.37iT - 71T^{2} \) |
| 73 | \( 1 + (3.67 - 2.12i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.51 - 7.81i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.55 + 7.55i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.03 - 1.75i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−13.40396265649831982980218836637, −12.94627390542883279849323823876, −11.66733910468485187830618758813, −9.850191785676275552912452579649, −8.818267395360592513371157944625, −7.949011449796339574401275325820, −6.91061057456994979933300522521, −5.96013198832079164294860383274, −4.13873251806264673359160340367, −1.18947487869370211986090565667,
3.20264783132959012111217307944, 3.54685214552715820463227541326, 5.51637846871348413645559429260, 7.47133586629831805951970842546, 8.877867525204984302157939808036, 9.772977886859533586585630001156, 10.50366690549670797826348725005, 11.31736137321288045841163390561, 12.59160320409587713515270220112, 13.76150589691038306223736870872