L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1 − 1.73i)5-s + 0.999·8-s + 1.99·10-s + (0.5 − 0.866i)11-s + (−3 − 5.19i)13-s + (−0.5 + 0.866i)16-s − 5·17-s + 7·19-s + (−0.999 + 1.73i)20-s + (0.499 + 0.866i)22-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + 6·26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.447 − 0.774i)5-s + 0.353·8-s + 0.632·10-s + (0.150 − 0.261i)11-s + (−0.832 − 1.44i)13-s + (−0.125 + 0.216i)16-s − 1.21·17-s + 1.60·19-s + (−0.223 + 0.387i)20-s + (0.106 + 0.184i)22-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + 1.17·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3197562297\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3197562297\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 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 | \( 1 - 2T + 37T^{2} \) |
| 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 | \( 1 + 12T + 53T^{2} \) |
| 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 | \( 1 + T + 73T^{2} \) |
| 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 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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
−8.482167070954187855283316261515, −7.64544554329000698447610735332, −7.33930056864690403149829449754, −6.19317360826032484676551970370, −5.31946472529751217489842616550, −4.88047222258114057166648986105, −3.78541404618012187678545809292, −2.72915644960097711265401061688, −1.21452442512922230404611134347, −0.12773854479166654269510290279,
1.59107382343939269847619266726, 2.57408066638272829838514521048, 3.39611634723352306465771727152, 4.36905592984673943753994428568, 5.02005326524542020550541820172, 6.43312372774243194834876687147, 7.05524617098335344162528340343, 7.56765898186647019957323494429, 8.597001314980560310076659636193, 9.389032723250591092267361799915