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 | \( 1 + 0.235T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.463790651661858693587499112867, −8.844237583519477008115139857408, −8.102045113268348745009792462309, −6.69412171885468812121411052678, −6.13339913883133018894914330936, −5.22560082078646023585598040844, −3.80520380917961074740535898863, −2.79192119173243052097677756663, −2.00158559117847632147920170559, −0.43610073958854642777460111858,
1.38942387423274938479790476512, 3.25349779085821202943161452644, 3.81326844092867257677584805104, 5.33923420065491607177834349899, 6.16444939309179017239986862573, 6.87247684752838497795780072168, 7.63920021122406549389006215500, 8.361626246796613598377482837073, 9.094141888330015058930398766942, 9.974653014107065854523456992615