L(s) = 1 | − 2.38·2-s + (1.37 − 2.38i)3-s + 3.70·4-s + (−0.491 + 0.850i)5-s + (−3.28 + 5.69i)6-s + (−0.911 − 2.48i)7-s − 4.06·8-s + (−2.28 − 3.95i)9-s + (1.17 − 2.03i)10-s + (0.293 − 0.509i)11-s + (5.09 − 8.82i)12-s + (2.39 − 2.69i)13-s + (2.17 + 5.93i)14-s + (1.35 + 2.34i)15-s + 2.30·16-s − 6.45·17-s + ⋯ |
L(s) = 1 | − 1.68·2-s + (0.794 − 1.37i)3-s + 1.85·4-s + (−0.219 + 0.380i)5-s + (−1.34 + 2.32i)6-s + (−0.344 − 0.938i)7-s − 1.43·8-s + (−0.761 − 1.31i)9-s + (0.370 − 0.642i)10-s + (0.0886 − 0.153i)11-s + (1.47 − 2.54i)12-s + (0.663 − 0.748i)13-s + (0.581 + 1.58i)14-s + (0.348 + 0.604i)15-s + 0.576·16-s − 1.56·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0420 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0420 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.387090 - 0.403726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.387090 - 0.403726i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.911 + 2.48i)T \) |
| 13 | \( 1 + (-2.39 + 2.69i)T \) |
good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 + (-1.37 + 2.38i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.491 - 0.850i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.293 + 0.509i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 6.45T + 17T^{2} \) |
| 19 | \( 1 + (-1.91 - 3.31i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 8.26T + 23T^{2} \) |
| 29 | \( 1 + (-1.98 - 3.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.49 - 2.58i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 + (1.83 + 3.17i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.19 - 5.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.17 + 3.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.212 + 0.368i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 6.00T + 59T^{2} \) |
| 61 | \( 1 + (1.10 + 1.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.50 - 6.07i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.80 - 3.11i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.46 + 4.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.39 - 2.41i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.86T + 83T^{2} \) |
| 89 | \( 1 + 2.09T + 89T^{2} \) |
| 97 | \( 1 + (3.84 - 6.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.63455827510047640744937011154, −12.88651406517246008285243002962, −11.30736858741022161026689467936, −10.47615387251609870953260983200, −9.042986889757231182177571062684, −8.242858608775841862859623617953, −7.19482148657063242044390288357, −6.69450342940659265769873821881, −3.03834500414905677830030191429, −1.20050738829685340082387120652,
2.64012804798322466310884390192, 4.58230946543916308580631230010, 6.70848710077142416325594022033, 8.453081858981078757719166546161, 8.974136893670771329719701190971, 9.530295632716137851986006853048, 10.76055129157585605263440856908, 11.61851090641497564946169244080, 13.40517790828564310338933454750, 15.03856824599576471663234253522