| L(s) = 1 | + (−1.26 − 0.636i)2-s + (2.73 − 0.685i)3-s + (1.18 + 1.60i)4-s + (−2.01 − 0.954i)5-s + (−3.89 − 0.877i)6-s + (1.40 + 4.64i)7-s + (−0.476 − 2.78i)8-s + (4.37 − 2.34i)9-s + (1.93 + 2.48i)10-s + (4.26 − 0.633i)11-s + (4.35 + 3.58i)12-s + (1.56 − 4.38i)13-s + (1.17 − 6.76i)14-s + (−6.17 − 1.22i)15-s + (−1.17 + 3.82i)16-s + (−2.34 + 0.467i)17-s + ⋯ |
| L(s) = 1 | + (−0.892 − 0.450i)2-s + (1.58 − 0.395i)3-s + (0.594 + 0.804i)4-s + (−0.902 − 0.426i)5-s + (−1.58 − 0.358i)6-s + (0.532 + 1.75i)7-s + (−0.168 − 0.985i)8-s + (1.45 − 0.780i)9-s + (0.613 + 0.787i)10-s + (1.28 − 0.190i)11-s + (1.25 + 1.03i)12-s + (0.434 − 1.21i)13-s + (0.315 − 1.80i)14-s + (−1.59 − 0.317i)15-s + (−0.293 + 0.955i)16-s + (−0.569 + 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27407 - 0.374210i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27407 - 0.374210i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 + 0.636i)T \) |
| good | 3 | \( 1 + (-2.73 + 0.685i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (2.01 + 0.954i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (-1.40 - 4.64i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-4.26 + 0.633i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-1.56 + 4.38i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (2.34 - 0.467i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-1.68 - 1.53i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (4.18 - 0.412i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (2.22 + 2.99i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (2.60 + 1.07i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.0217 - 0.443i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (1.25 - 1.52i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (0.294 - 1.17i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-7.81 - 5.22i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (7.94 - 10.7i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (1.75 + 4.91i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (0.0389 - 0.0649i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (7.39 + 4.43i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-2.73 + 5.11i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-3.39 + 11.1i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-1.40 - 2.10i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.0674 - 1.37i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (3.87 + 0.381i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (5.33 - 12.8i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−12.11292041550520167467877385319, −11.08857452396110256393833443489, −9.426791769159994240159392613828, −8.969062521636429126703791996423, −8.111086826439752370999800384943, −7.81969140807663852579234034572, −6.12796869150614390832068183580, −3.98723277742078804738777908740, −2.91959865706607269340759497760, −1.69851492635886136362650518847,
1.69203318851196452546164258386, 3.68368068983567535176040282939, 4.31050339108121745247787391330, 6.90556243157093863888706508606, 7.26562136181753379475908881470, 8.261211046431460205106240126126, 9.070136013572723999407772424962, 9.908338467472658990945804707816, 10.95976978913571220573671142358, 11.63127469741912759515404203180
