| L(s) = 1 | + (0.450 − 1.34i)2-s + (0.195 + 1.36i)3-s + (−1.59 − 1.20i)4-s + (−2.10 + 0.758i)5-s + (1.91 + 0.351i)6-s + (2.25 + 1.95i)7-s + (−2.33 + 1.59i)8-s + (1.06 − 0.313i)9-s + (0.0681 + 3.16i)10-s + (−1.97 + 1.27i)11-s + (1.33 − 2.40i)12-s + (−1.81 + 1.57i)13-s + (3.63 − 2.14i)14-s + (−1.44 − 2.71i)15-s + (1.07 + 3.85i)16-s + (−2.71 + 5.95i)17-s + ⋯ |
| L(s) = 1 | + (0.318 − 0.947i)2-s + (0.112 + 0.785i)3-s + (−0.796 − 0.604i)4-s + (−0.940 + 0.339i)5-s + (0.780 + 0.143i)6-s + (0.851 + 0.738i)7-s + (−0.826 + 0.562i)8-s + (0.355 − 0.104i)9-s + (0.0215 + 0.999i)10-s + (−0.596 + 0.383i)11-s + (0.384 − 0.693i)12-s + (−0.503 + 0.436i)13-s + (0.971 − 0.572i)14-s + (−0.372 − 0.700i)15-s + (0.269 + 0.963i)16-s + (−0.659 + 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08552 + 0.530087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08552 + 0.530087i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.450 + 1.34i)T \) |
| 5 | \( 1 + (2.10 - 0.758i)T \) |
| 23 | \( 1 + (0.0393 - 4.79i)T \) |
| good | 3 | \( 1 + (-0.195 - 1.36i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-2.25 - 1.95i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (1.97 - 1.27i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.81 - 1.57i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.71 - 5.95i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.209 - 0.458i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.03 + 6.63i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-8.59 - 1.23i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-4.42 + 1.30i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.227 + 0.0668i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (11.1 - 1.59i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 8.41T + 47T^{2} \) |
| 53 | \( 1 + (-2.79 + 3.22i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (8.81 - 7.63i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (7.90 + 1.13i)T + (58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.03 + 6.27i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-2.76 + 4.29i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-8.40 + 3.83i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.508 - 0.587i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (0.865 + 2.94i)T + (-69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (3.79 - 0.545i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (2.41 + 0.708i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.26298752972175920604912247499, −10.36307568001278945132422539190, −9.717214840473925950265255130386, −8.588261678353952115538728293638, −7.889162943779614814090448424028, −6.32999669373549913561405058845, −4.85620951638532304318325499868, −4.41553227242532998798316316756, −3.28898667029919724689725193891, −1.94804228354552312803481280633,
0.69651063218757950387426449388, 2.97109461174962191771661639918, 4.61646288409453695526706203342, 4.86841162024691397227132029922, 6.63485342644341555548244134372, 7.28670306286605885453886647953, 8.020267420647540234005920705713, 8.509805652962501749307923270235, 9.938276822204395460829003122098, 11.13267250978966894848839800270
