| L(s) = 1 | + (0.766 + 0.642i)3-s + (−2.67 + 0.972i)5-s + (−2.07 + 1.20i)7-s + (0.173 + 0.984i)9-s + (0.768 + 0.443i)11-s + (−1.53 − 1.82i)13-s + (−2.67 − 0.972i)15-s + (0.112 − 0.638i)17-s + (−0.564 − 4.32i)19-s + (−2.36 − 0.416i)21-s + (1.35 − 3.70i)23-s + (2.36 − 1.98i)25-s + (−0.500 + 0.866i)27-s + (−6.77 + 1.19i)29-s + (−3.97 − 6.88i)31-s + ⋯ |
| L(s) = 1 | + (0.442 + 0.371i)3-s + (−1.19 + 0.434i)5-s + (−0.785 + 0.453i)7-s + (0.0578 + 0.328i)9-s + (0.231 + 0.133i)11-s + (−0.424 − 0.506i)13-s + (−0.689 − 0.251i)15-s + (0.0273 − 0.154i)17-s + (−0.129 − 0.991i)19-s + (−0.515 − 0.0909i)21-s + (0.281 − 0.773i)23-s + (0.473 − 0.396i)25-s + (−0.0962 + 0.166i)27-s + (−1.25 + 0.221i)29-s + (−0.714 − 1.23i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0480137 - 0.112807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0480137 - 0.112807i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (0.564 + 4.32i)T \) |
| good | 5 | \( 1 + (2.67 - 0.972i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (2.07 - 1.20i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.768 - 0.443i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.53 + 1.82i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.112 + 0.638i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.35 + 3.70i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (6.77 - 1.19i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (3.97 + 6.88i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.18iT - 37T^{2} \) |
| 41 | \( 1 + (4.03 - 4.81i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.15 + 5.92i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.153 - 0.0271i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (4.67 - 12.8i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.401 + 2.27i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (7.99 + 2.91i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.43 + 8.16i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.13 + 0.776i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.89 - 7.45i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-9.24 - 7.76i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (8.09 - 4.67i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.44 - 10.0i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.73 + 0.481i)T + (91.1 + 33.1i)T^{2} \) |
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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
−9.555576493300791517829850774194, −9.118707937462949577773963319470, −8.014509254327961995618384574160, −7.36971192399484086141606890680, −6.49304355133945174495983974150, −5.28014443361286167784353325912, −4.17652617240789227276105608333, −3.35767095999276679463556390651, −2.47179944082414031998940084817, −0.05360058802825877995609596890,
1.62601474052134902292912931989, 3.37118489739222545390379478048, 3.82610638646861065801656217038, 5.01210922606386679560551512985, 6.32269038612007502143892072372, 7.18836617732352577194261604265, 7.81953368671743327382791180094, 8.657992889665570306738303680676, 9.460771746758874014170351112073, 10.30188345374514226811027059895
