L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.707 + 1.22i)5-s + 0.999·8-s + (−0.707 − 1.22i)10-s + (−2 − 3.46i)11-s + 4.24·13-s + (−0.5 + 0.866i)16-s + (3.53 + 6.12i)17-s + (2.82 − 4.89i)19-s + 1.41·20-s + 3.99·22-s + (−4 + 6.92i)23-s + (1.50 + 2.59i)25-s + (−2.12 + 3.67i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.316 + 0.547i)5-s + 0.353·8-s + (−0.223 − 0.387i)10-s + (−0.603 − 1.04i)11-s + 1.17·13-s + (−0.125 + 0.216i)16-s + (0.857 + 1.48i)17-s + (0.648 − 1.12i)19-s + 0.316·20-s + 0.852·22-s + (−0.834 + 1.44i)23-s + (0.300 + 0.519i)25-s + (−0.416 + 0.720i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.850647 + 0.791007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.850647 + 0.791007i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.707 - 1.22i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + (-3.53 - 6.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.82 + 4.89i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (2.82 - 4.89i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.65 - 9.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.707 - 1.22i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-7.77 - 13.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + (3.53 - 6.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−10.35730984445575302873164352158, −9.403891749094703590082938870962, −8.412960019789742293565165114477, −7.927136793112660817995535839871, −6.97637096296907818112749860725, −5.99002442487169939308747345805, −5.43850829454985606710629406143, −3.91256610908910115808884881165, −3.07946469170882912184006276212, −1.21843116791265542729606559891,
0.76092945166505975235038241402, 2.18695991845347059750656999089, 3.43221590736018038087150992813, 4.46517422454321559263041706085, 5.33030396604507299284561070714, 6.58300978484820659563953965345, 7.76397569064066064729458178448, 8.196907431428960743941237319237, 9.238323523012022438378181029489, 9.965630761103930081025017570976