L(s) = 1 | + (−0.202 + 0.350i)2-s − 0.0458i·3-s + (0.917 + 1.58i)4-s + (0.0766 − 0.286i)5-s + (0.0160 + 0.00928i)6-s + (0.00171 + 0.00171i)7-s − 1.55·8-s + 2.99·9-s + (0.0848 + 0.0848i)10-s + (−3.03 − 0.813i)11-s + (0.0729 − 0.0421i)12-s + (−0.799 − 2.98i)13-s + (−0.000947 + 0.000253i)14-s + (−0.0131 − 0.00351i)15-s + (−1.52 + 2.63i)16-s + (−4.10 − 4.10i)17-s + ⋯ |
L(s) = 1 | + (−0.143 + 0.248i)2-s − 0.0264i·3-s + (0.458 + 0.794i)4-s + (0.0342 − 0.127i)5-s + (0.00656 + 0.00379i)6-s + (0.000646 + 0.000646i)7-s − 0.549·8-s + 0.999·9-s + (0.0268 + 0.0268i)10-s + (−0.915 − 0.245i)11-s + (0.0210 − 0.0121i)12-s + (−0.221 − 0.827i)13-s + (−0.000253 + 6.78e−5i)14-s + (−0.00338 − 0.000908i)15-s + (−0.380 + 0.658i)16-s + (−0.995 − 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.885503 + 0.279074i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.885503 + 0.279074i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 + (8.54 - 0.245i)T \) |
good | 2 | \( 1 + (0.202 - 0.350i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + 0.0458iT - 3T^{2} \) |
| 5 | \( 1 + (-0.0766 + 0.286i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (-0.00171 - 0.00171i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.03 + 0.813i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (0.799 + 2.98i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (4.10 + 4.10i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.16 + 1.24i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.40 + 0.813i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.129 + 0.483i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-1.37 - 5.13i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-2.56 - 4.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.48 - 6.03i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.55 - 3.55i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.430 + 0.115i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.36 + 1.17i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (8.29 - 2.22i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-6.08 + 3.51i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.01 + 5.20i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.21 - 12.5i)T + (-35.5 - 61.4i)T^{2} \) |
| 79 | \( 1 + (-7.02 - 4.05i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.97 - 2.97i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.78 + 6.55i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.25iT - 97T^{2} \) |
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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
−15.08678371581629773495316419952, −13.25409528197398088836610181519, −12.78944257898999819981751912490, −11.44093392517976876862571410116, −10.30545410913289068933036452693, −8.822171998713283276229289852052, −7.66677372245933987463866361499, −6.67512775960394561833258482645, −4.80360513681078546836717064781, −2.85938124396687415927740767820,
2.11347276926257703104396827397, 4.50647084234277519292556653610, 6.17131293442517796760325875810, 7.31969323566633786677255560950, 9.044585250300658395050384593495, 10.25856618060113264268647267983, 10.89942187160837549189203875252, 12.30059526688830333659377608690, 13.36077383814855775239427636127, 14.73534012136085947595158631477