L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.415 + 0.909i)3-s + (−0.959 + 0.281i)4-s + (−1.81 + 2.09i)5-s + (−0.959 − 0.281i)6-s + (0.163 − 0.105i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (−2.33 − 1.49i)10-s + (−0.413 + 2.87i)11-s + (0.142 − 0.989i)12-s + (0.910 + 0.585i)13-s + (0.127 + 0.146i)14-s + (−1.15 − 2.52i)15-s + (0.841 − 0.540i)16-s + (4.45 + 1.30i)17-s + ⋯ |
L(s) = 1 | + (0.100 + 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (−0.811 + 0.936i)5-s + (−0.391 − 0.115i)6-s + (0.0617 − 0.0396i)7-s + (−0.146 − 0.321i)8-s + (−0.218 − 0.251i)9-s + (−0.737 − 0.473i)10-s + (−0.124 + 0.866i)11-s + (0.0410 − 0.285i)12-s + (0.252 + 0.162i)13-s + (0.0339 + 0.0392i)14-s + (−0.297 − 0.650i)15-s + (0.210 − 0.135i)16-s + (1.08 + 0.317i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.310919 + 0.807764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.310919 + 0.807764i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (4.73 - 0.757i)T \) |
good | 5 | \( 1 + (1.81 - 2.09i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.163 + 0.105i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.413 - 2.87i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.910 - 0.585i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.45 - 1.30i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.24 + 1.24i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-8.83 - 2.59i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.85 + 6.25i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-3.17 - 3.66i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.76 + 4.34i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.14 + 6.88i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 5.61T + 47T^{2} \) |
| 53 | \( 1 + (3.48 - 2.23i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (9.01 + 5.79i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (1.19 + 2.61i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.08 - 14.4i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.81 + 12.6i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.86 - 0.841i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-2.15 - 1.38i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-0.531 - 0.613i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.81 + 6.16i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-8.09 + 9.34i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−13.97646676471361403578080768752, −12.44854416126228092282495356959, −11.57147006560804487326066538626, −10.44103685618951582477742888137, −9.519218077804436595782125333060, −7.996802543237717244077371433565, −7.21219480572979097330422854589, −5.96494471133739929134685424002, −4.55348943232094850890617485398, −3.33733789944420783868441498755,
0.980856827806809260870526227219, 3.25861174126809108792451682236, 4.75495145381039917345914694195, 5.95555070429888273802535979893, 7.79592252407485357542558939765, 8.453932190353415760640027169907, 9.786693720069600306852538048255, 11.06818653510846912104025811763, 11.99487199290513926926868916920, 12.46572957227381424076420600210