L(s) = 1 | + (0.5 − 0.866i)2-s − 1.73i·3-s + (−0.499 − 0.866i)4-s − 2·5-s + (−1.49 − 0.866i)6-s − 0.999·8-s − 2.99·9-s + (−1 + 1.73i)10-s + 11-s + (−1.49 + 0.866i)12-s + (−3 + 5.19i)13-s + 3.46i·15-s + (−0.5 + 0.866i)16-s + (−2.5 + 4.33i)17-s + (−1.49 + 2.59i)18-s + (−3.5 − 6.06i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s − 0.999i·3-s + (−0.249 − 0.433i)4-s − 0.894·5-s + (−0.612 − 0.353i)6-s − 0.353·8-s − 0.999·9-s + (−0.316 + 0.547i)10-s + 0.301·11-s + (−0.433 + 0.249i)12-s + (−0.832 + 1.44i)13-s + 0.894i·15-s + (−0.125 + 0.216i)16-s + (−0.606 + 1.05i)17-s + (−0.353 + 0.612i)18-s + (−0.802 − 1.39i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0788 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0788 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6 - 10.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8 - 13.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.232718206710121593931840859865, −8.834762969766625512755992773921, −7.66789065723497343745787176113, −6.90918376905108644980201328654, −6.17537932399537230252458479174, −4.77276804566385830257117393465, −4.03053458635514802427630038701, −2.71572073857707648349668193531, −1.70077269937731846127516282317, 0,
2.84575367638246797826555625003, 3.71174488859932415149020344677, 4.63919110831609025510858491016, 5.33373518892321937314664540923, 6.38357716082754974938204037881, 7.53330820577589880628333198545, 8.153032747296355867613584829145, 8.997725958452179417790194004687, 9.940396021989886035491900777975