L(s) = 1 | + (−0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.884 − 1.21i)5-s + (0.809 − 0.587i)6-s + (−1.60 + 2.10i)7-s + (0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + 1.50·10-s + (2.41 + 2.27i)11-s + i·12-s + (−2.55 − 1.85i)13-s + (−0.763 − 2.53i)14-s + (0.465 + 1.43i)15-s + (−0.809 + 0.587i)16-s + (3.63 − 2.64i)17-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.572i)2-s + (−0.549 − 0.178i)3-s + (−0.154 − 0.475i)4-s + (−0.395 − 0.544i)5-s + (0.330 − 0.239i)6-s + (−0.604 + 0.796i)7-s + (0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s + 0.475·10-s + (0.727 + 0.685i)11-s + 0.288i·12-s + (−0.707 − 0.514i)13-s + (−0.204 − 0.677i)14-s + (0.120 + 0.369i)15-s + (−0.202 + 0.146i)16-s + (0.882 − 0.641i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.793726 - 0.0868776i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.793726 - 0.0868776i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 3 | \( 1 + (0.951 + 0.309i)T \) |
| 7 | \( 1 + (1.60 - 2.10i)T \) |
| 11 | \( 1 + (-2.41 - 2.27i)T \) |
good | 5 | \( 1 + (0.884 + 1.21i)T + (-1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (2.55 + 1.85i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.63 + 2.64i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.96 + 6.04i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 + (-0.814 + 0.264i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.69 + 5.08i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.85 + 5.72i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (3.11 - 9.59i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.65iT - 43T^{2} \) |
| 47 | \( 1 + (-1.41 - 0.460i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.331 - 0.240i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-5.04 + 1.63i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.80 + 4.21i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + (-11.6 + 8.43i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.34 - 13.3i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.69 + 10.5i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (2.08 - 1.51i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 2.90iT - 89T^{2} \) |
| 97 | \( 1 + (9.52 - 13.1i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.11084344322307515038431504204, −9.701282837276893508443606484928, −9.392528287862991152813457880545, −8.257446107208586796089706287922, −7.23262789473257136135059445453, −6.53278558870682895344238587697, −5.31389714118262674017550335303, −4.67127331758322187712216563532, −2.80852033275305422440789082485, −0.77361255874662371745337850754,
1.15607180551800653789478427842, 3.25546516003530730640721503688, 3.87352772185538460325763782320, 5.33826881122325522614022159370, 6.70492651976684320441854261552, 7.26697020501013773751702262696, 8.503722849150994778789662761177, 9.560991362407263656031938362954, 10.36057418286913064760920099571, 10.89631554070534220484324745947