L(s) = 1 | + (0.5 + 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 + 0.866i)4-s − 1.99·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.999 − 1.73i)12-s + 4·13-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (0.499 − 0.866i)18-s + (1 + 1.73i)19-s + (0.999 − 1.73i)24-s + (2.5 − 4.33i)25-s + (2 + 3.46i)26-s − 4.00·27-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s − 0.816·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.288 − 0.499i)12-s + 1.10·13-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (0.117 − 0.204i)18-s + (0.229 + 0.397i)19-s + (0.204 − 0.353i)24-s + (0.5 − 0.866i)25-s + (0.392 + 0.679i)26-s − 0.769·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.606661 + 0.797446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.606661 + 0.797446i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−14.35056284651231564581666825333, −13.39276982253282504579114951402, −12.06926607584256368231314481450, −11.03156301358370933522671055749, −9.969986304007682103457348749805, −8.801779702387547062476919061217, −7.38187572562548851992014849471, −5.90847417571958222976833557124, −4.92836069455357757789022627749, −3.60730141195257487979543238621,
1.47640700840045322049156739130, 3.65733990102187077391429675053, 5.54613123540807352222510231172, 6.53895455765916395067666626447, 7.937843169684972630689982770456, 9.390512869525057287713939002024, 10.83699373690739446285393590903, 11.55908781045891273671686991799, 12.75709997600820319402207612824, 13.12545459994660673294455378565