L(s) = 1 | + (−0.703 + 1.21i)2-s + (0.0108 + 0.0188i)4-s + (0.147 − 0.256i)5-s + (−1.43 − 2.22i)7-s − 2.84·8-s + (0.207 + 0.360i)10-s + (−0.186 − 0.322i)11-s + 3.46·13-s + (3.71 − 0.183i)14-s + (1.97 − 3.42i)16-s + (1.64 + 2.85i)17-s + (−0.5 + 0.866i)19-s + 0.00643·20-s + 0.524·22-s + (−3.70 + 6.41i)23-s + ⋯ |
L(s) = 1 | + (−0.497 + 0.861i)2-s + (0.00543 + 0.00941i)4-s + (0.0661 − 0.114i)5-s + (−0.542 − 0.840i)7-s − 1.00·8-s + (0.0657 + 0.113i)10-s + (−0.0561 − 0.0972i)11-s + 0.962·13-s + (0.993 − 0.0489i)14-s + (0.494 − 0.856i)16-s + (0.399 + 0.692i)17-s + (−0.114 + 0.198i)19-s + 0.00143·20-s + 0.111·22-s + (−0.772 + 1.33i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8422349476\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8422349476\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.43 + 2.22i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.703 - 1.21i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.147 + 0.256i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.186 + 0.322i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + (-1.64 - 2.85i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.70 - 6.41i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + (-1.98 - 3.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.91 - 8.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 + 6.08T + 43T^{2} \) |
| 47 | \( 1 + (-3.41 + 5.90i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.21 - 9.03i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.34 + 4.05i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.54 - 6.14i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.83 - 6.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.88T + 71T^{2} \) |
| 73 | \( 1 + (-0.508 - 0.881i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.30 + 9.18i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + (3.09 - 5.35i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.235T + 97T^{2} \) |
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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
−9.974653014107065854523456992615, −9.094141888330015058930398766942, −8.361626246796613598377482837073, −7.63920021122406549389006215500, −6.87247684752838497795780072168, −6.16444939309179017239986862573, −5.33923420065491607177834349899, −3.81326844092867257677584805104, −3.25349779085821202943161452644, −1.38942387423274938479790476512,
0.43610073958854642777460111858, 2.00158559117847632147920170559, 2.79192119173243052097677756663, 3.80520380917961074740535898863, 5.22560082078646023585598040844, 6.13339913883133018894914330936, 6.69412171885468812121411052678, 8.102045113268348745009792462309, 8.844237583519477008115139857408, 9.463790651661858693587499112867