L(s) = 1 | + (−1.64 + 0.375i)3-s + (−0.945 − 1.18i)5-s + (0.445 + 1.95i)7-s + (−0.132 + 0.0637i)9-s + (−2.56 + 5.32i)11-s + (−3.47 − 1.67i)13-s + (2.00 + 1.59i)15-s + 2.92i·17-s + (−4.67 − 1.06i)19-s + (−1.46 − 3.04i)21-s + (−2.78 + 3.49i)23-s + (0.600 − 2.63i)25-s + (4.15 − 3.31i)27-s + (4.43 + 3.05i)29-s + (1.99 − 1.59i)31-s + ⋯ |
L(s) = 1 | + (−0.950 + 0.217i)3-s + (−0.423 − 0.530i)5-s + (0.168 + 0.737i)7-s + (−0.0441 + 0.0212i)9-s + (−0.772 + 1.60i)11-s + (−0.963 − 0.463i)13-s + (0.517 + 0.412i)15-s + 0.709i·17-s + (−1.07 − 0.244i)19-s + (−0.320 − 0.664i)21-s + (−0.581 + 0.728i)23-s + (0.120 − 0.526i)25-s + (0.799 − 0.637i)27-s + (0.823 + 0.566i)29-s + (0.359 − 0.286i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.125136 + 0.350464i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.125136 + 0.350464i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + (-4.43 - 3.05i)T \) |
good | 3 | \( 1 + (1.64 - 0.375i)T + (2.70 - 1.30i)T^{2} \) |
| 5 | \( 1 + (0.945 + 1.18i)T + (-1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (-0.445 - 1.95i)T + (-6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (2.56 - 5.32i)T + (-6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (3.47 + 1.67i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 - 2.92iT - 17T^{2} \) |
| 19 | \( 1 + (4.67 + 1.06i)T + (17.1 + 8.24i)T^{2} \) |
| 23 | \( 1 + (2.78 - 3.49i)T + (-5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (-1.99 + 1.59i)T + (6.89 - 30.2i)T^{2} \) |
| 37 | \( 1 + (-0.355 - 0.737i)T + (-23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 - 8.28iT - 41T^{2} \) |
| 43 | \( 1 + (1.04 + 0.830i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-2.15 + 4.46i)T + (-29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (1.94 + 2.44i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + 7.30T + 59T^{2} \) |
| 61 | \( 1 + (-3.52 + 0.805i)T + (54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + (5.55 - 2.67i)T + (41.7 - 52.3i)T^{2} \) |
| 71 | \( 1 + (1.63 + 0.788i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (8.52 + 6.79i)T + (16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 + (-6.17 - 12.8i)T + (-49.2 + 61.7i)T^{2} \) |
| 83 | \( 1 + (-3.25 + 14.2i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (9.99 - 7.97i)T + (19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + (-8.03 - 1.83i)T + (87.3 + 42.0i)T^{2} \) |
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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
−12.27855448947714581490473125346, −11.89317169099747481786442601150, −10.59215864073392580230912373339, −9.932154124136718269547790904830, −8.566024595545531493284931942520, −7.67406749725790406398804963149, −6.30638444303617807200367550982, −5.12289460151530795773469544730, −4.54450959968060560617472778785, −2.35535434641258422706385710727,
0.32231055254011606618228058711, 2.90698046601660273433026518906, 4.44744848739027747723118561548, 5.71623912837119737656737249287, 6.67814336543083397223932700694, 7.66260064896716361034008199588, 8.757186837751782854717995414188, 10.34336299802810659422937347414, 10.90921591492676205106940972479, 11.67493168623153914900304420037