L(s) = 1 | + (0.679 + 1.59i)3-s + (−2.39 − 0.642i)5-s + (1.93 − 3.34i)7-s + (−2.07 + 2.16i)9-s + (4.01 − 1.07i)11-s + (3.17 + 0.850i)13-s + (−0.605 − 4.25i)15-s − 1.33i·17-s + (6.09 + 6.09i)19-s + (6.64 + 0.804i)21-s + (−0.521 + 0.301i)23-s + (1.00 + 0.582i)25-s + (−4.86 − 1.83i)27-s + (0.272 − 0.0730i)29-s + (5.84 − 3.37i)31-s + ⋯ |
L(s) = 1 | + (0.392 + 0.919i)3-s + (−1.07 − 0.287i)5-s + (0.730 − 1.26i)7-s + (−0.692 + 0.721i)9-s + (1.21 − 0.324i)11-s + (0.880 + 0.235i)13-s + (−0.156 − 1.09i)15-s − 0.322i·17-s + (1.39 + 1.39i)19-s + (1.44 + 0.175i)21-s + (−0.108 + 0.0627i)23-s + (0.201 + 0.116i)25-s + (−0.935 − 0.353i)27-s + (0.0506 − 0.0135i)29-s + (1.05 − 0.606i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60216 + 0.236414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60216 + 0.236414i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.679 - 1.59i)T \) |
good | 5 | \( 1 + (2.39 + 0.642i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-1.93 + 3.34i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.01 + 1.07i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-3.17 - 0.850i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + 1.33iT - 17T^{2} \) |
| 19 | \( 1 + (-6.09 - 6.09i)T + 19iT^{2} \) |
| 23 | \( 1 + (0.521 - 0.301i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.272 + 0.0730i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-5.84 + 3.37i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.00346 - 0.00346i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.614 + 1.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.151 - 0.563i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (1.24 - 2.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.24 - 3.24i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.44 - 5.40i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.528 - 1.97i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-2.65 + 9.90i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 7.85iT - 71T^{2} \) |
| 73 | \( 1 + 7.41iT - 73T^{2} \) |
| 79 | \( 1 + (-0.839 - 0.484i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.171 - 0.639i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + (3.24 - 5.62i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.82035836856541815394521965621, −9.902440762879705077302151755856, −8.996153054198399597890323159157, −8.027009972137234972197625192601, −7.60438547400940106441971746024, −6.17003366825834789475125850815, −4.78703603861844668763246350786, −3.96657778097681038052164569850, −3.50885775322672942137886896805, −1.20024655179905370609017981934,
1.29750734245311817344825205439, 2.73625678118323298629716249509, 3.78607812918466294041793638377, 5.17534936075910723678528841266, 6.35619537552972450893109081256, 7.15086765407713513515803356951, 8.132791543857568200666502781488, 8.666674614801813744661441756028, 9.491136348583211828581753569523, 11.16431252185256621934934297054