L(s) = 1 | + (−0.0500 + 0.127i)2-s + (−1.68 + 0.411i)3-s + (1.45 + 1.34i)4-s + (2.28 − 1.09i)5-s + (0.0317 − 0.234i)6-s + (2.44 + 1.01i)7-s + (−0.491 + 0.236i)8-s + (2.66 − 1.38i)9-s + (0.0259 + 0.345i)10-s + (−2.32 − 2.90i)11-s + (−2.99 − 1.66i)12-s + (0.926 − 2.36i)13-s + (−0.251 + 0.260i)14-s + (−3.38 + 2.78i)15-s + (0.290 + 3.87i)16-s + (3.88 − 3.60i)17-s + ⋯ |
L(s) = 1 | + (−0.0353 + 0.0901i)2-s + (−0.971 + 0.237i)3-s + (0.726 + 0.673i)4-s + (1.02 − 0.491i)5-s + (0.0129 − 0.0959i)6-s + (0.922 + 0.385i)7-s + (−0.173 + 0.0836i)8-s + (0.887 − 0.461i)9-s + (0.00819 + 0.109i)10-s + (−0.699 − 0.877i)11-s + (−0.865 − 0.482i)12-s + (0.257 − 0.654i)13-s + (−0.0673 + 0.0695i)14-s + (−0.874 + 0.719i)15-s + (0.0726 + 0.969i)16-s + (0.943 − 0.875i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49009 + 0.353121i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49009 + 0.353121i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.68 - 0.411i)T \) |
| 7 | \( 1 + (-2.44 - 1.01i)T \) |
good | 2 | \( 1 + (0.0500 - 0.127i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.28 + 1.09i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (2.32 + 2.90i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.926 + 2.36i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-3.88 + 3.60i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.59 - 4.49i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.816 - 3.57i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-2.55 - 0.788i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-5.40 + 9.35i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.32 - 1.64i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-2.77 - 1.89i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (8.66 - 5.90i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (1.12 - 2.87i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (12.3 - 3.82i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-6.38 + 4.35i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (6.13 - 5.69i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (4.21 - 7.29i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.21 + 9.68i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (8.46 + 1.27i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (0.0538 + 0.0932i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.62 + 11.7i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (4.55 + 11.6i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-7.56 + 13.0i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.34705420820434384364821697292, −10.39714549376624786599096667488, −9.572693277458208465733092298352, −8.226881935069106449033354323860, −7.67172458795550842500712039316, −5.97756173399175661854040419672, −5.83903033204675051655942637656, −4.64185529248964767645980960254, −3.00466173007294673968387341869, −1.47139597408839787811187928800,
1.43790660847704036641824868817, 2.34662914447197479390671801205, 4.60740466565845877550623024732, 5.41807209457562938906425134569, 6.46963013652599087806818635760, 6.92909669409781455851377322190, 8.140804190702682551987208770357, 9.760125390982870154861458052075, 10.46047498138206070983344457363, 10.78148024815514580068683491591