| L(s) = 1 | + (−0.928 − 0.371i)2-s + (−0.184 − 1.93i)3-s + (0.723 + 0.690i)4-s + (0.841 + 2.43i)5-s + (−0.547 + 1.86i)6-s + (−0.971 − 2.46i)7-s + (−0.415 − 0.909i)8-s + (−0.760 + 0.146i)9-s + (0.122 − 2.57i)10-s + (−0.422 − 1.05i)11-s + (1.20 − 1.52i)12-s + (3.20 − 4.97i)13-s + (−0.0131 + 2.64i)14-s + (4.54 − 2.07i)15-s + (0.0475 + 0.998i)16-s + (0.280 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.656 − 0.262i)2-s + (−0.106 − 1.11i)3-s + (0.361 + 0.345i)4-s + (0.376 + 1.08i)5-s + (−0.223 + 0.761i)6-s + (−0.367 − 0.930i)7-s + (−0.146 − 0.321i)8-s + (−0.253 + 0.0488i)9-s + (0.0387 − 0.812i)10-s + (−0.127 − 0.318i)11-s + (0.346 − 0.440i)12-s + (0.887 − 1.38i)13-s + (−0.00352 + 0.707i)14-s + (1.17 − 0.536i)15-s + (0.0118 + 0.249i)16-s + (0.0679 + 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.583678 - 0.738928i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583678 - 0.738928i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.928 + 0.371i)T \) |
| 7 | \( 1 + (0.971 + 2.46i)T \) |
| 23 | \( 1 + (1.25 + 4.62i)T \) |
| good | 3 | \( 1 + (0.184 + 1.93i)T + (-2.94 + 0.567i)T^{2} \) |
| 5 | \( 1 + (-0.841 - 2.43i)T + (-3.93 + 3.09i)T^{2} \) |
| 11 | \( 1 + (0.422 + 1.05i)T + (-7.96 + 7.59i)T^{2} \) |
| 13 | \( 1 + (-3.20 + 4.97i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.280 - 1.15i)T + (-15.1 + 7.78i)T^{2} \) |
| 19 | \( 1 + (0.0599 - 0.246i)T + (-16.8 - 8.70i)T^{2} \) |
| 29 | \( 1 + (1.05 + 0.309i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (5.74 + 4.09i)T + (10.1 + 29.2i)T^{2} \) |
| 37 | \( 1 + (-0.843 - 4.37i)T + (-34.3 + 13.7i)T^{2} \) |
| 41 | \( 1 + (-3.72 - 3.22i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.04 + 1.38i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-10.2 - 5.92i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.684 - 1.32i)T + (-30.7 - 43.1i)T^{2} \) |
| 59 | \( 1 + (-4.35 - 0.207i)T + (58.7 + 5.60i)T^{2} \) |
| 61 | \( 1 + (-4.22 - 0.403i)T + (59.8 + 11.5i)T^{2} \) |
| 67 | \( 1 + (5.00 + 6.36i)T + (-15.7 + 65.1i)T^{2} \) |
| 71 | \( 1 + (-1.37 - 9.54i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-10.1 + 10.6i)T + (-3.47 - 72.9i)T^{2} \) |
| 79 | \( 1 + (-5.02 - 9.74i)T + (-45.8 + 64.3i)T^{2} \) |
| 83 | \( 1 + (-11.3 - 13.1i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (6.25 + 8.78i)T + (-29.1 + 84.1i)T^{2} \) |
| 97 | \( 1 + (5.50 - 6.34i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.97503081502303260296875997413, −10.63192854511131231165422898306, −9.720286403875016677189150859811, −8.228866912472279002145153370250, −7.54821881192984787698295907288, −6.63920276906826281974835079056, −5.98645951350724445594095599325, −3.69993894081171044166946327362, −2.51700227454967728253509107083, −0.898305390429527360519385924684,
1.81872511569779952639186985335, 3.82613637506146074557824038298, 5.05701178189247231593259288042, 5.79526231341044746764571333739, 7.12142555247486214879317008969, 8.649253706052121060538638053989, 9.189775190909403369472414633820, 9.633289144824320028507954006606, 10.77810568148681601317551497578, 11.70695009624146695439804020617
