L(s) = 1 | + (−2.42 − 1.00i)5-s + (−3.40 − 3.40i)7-s + (0.847 − 2.04i)11-s + (−1.82 + 0.757i)13-s + 7.04i·17-s + (4.96 − 2.05i)19-s + (−3.37 + 3.37i)23-s + (1.32 + 1.32i)25-s + (2.83 + 6.84i)29-s − 1.94·31-s + (4.83 + 11.6i)35-s + (4.02 + 1.66i)37-s + (−0.970 + 0.970i)41-s + (0.0467 − 0.112i)43-s + 6.49i·47-s + ⋯ |
L(s) = 1 | + (−1.08 − 0.448i)5-s + (−1.28 − 1.28i)7-s + (0.255 − 0.616i)11-s + (−0.507 + 0.210i)13-s + 1.70i·17-s + (1.14 − 0.472i)19-s + (−0.703 + 0.703i)23-s + (0.265 + 0.265i)25-s + (0.526 + 1.27i)29-s − 0.348·31-s + (0.817 + 1.97i)35-s + (0.661 + 0.274i)37-s + (−0.151 + 0.151i)41-s + (0.00712 − 0.0172i)43-s + 0.946i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3305968346\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3305968346\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.42 + 1.00i)T + (3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (3.40 + 3.40i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.847 + 2.04i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (1.82 - 0.757i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 7.04iT - 17T^{2} \) |
| 19 | \( 1 + (-4.96 + 2.05i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (3.37 - 3.37i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.83 - 6.84i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 1.94T + 31T^{2} \) |
| 37 | \( 1 + (-4.02 - 1.66i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (0.970 - 0.970i)T - 41iT^{2} \) |
| 43 | \( 1 + (-0.0467 + 0.112i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 6.49iT - 47T^{2} \) |
| 53 | \( 1 + (-0.894 + 2.15i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (7.71 + 3.19i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (4.20 + 10.1i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (0.933 + 2.25i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-4.81 - 4.81i)T + 71iT^{2} \) |
| 73 | \( 1 + (2.31 - 2.31i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.41iT - 79T^{2} \) |
| 83 | \( 1 + (6.76 - 2.80i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (8.37 + 8.37i)T + 89iT^{2} \) |
| 97 | \( 1 + 5.80T + 97T^{2} \) |
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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
−9.986314740791134887772629350119, −9.289852821006928237099730840571, −8.253692251814379947365137092529, −7.57556041635612966156379755296, −6.79994959468613652342231827956, −5.94909727404852333444340442741, −4.62047461825668376466824112934, −3.76574067161382498059343701704, −3.24736427980775406082939502226, −1.15795414075531951687755832576,
0.16270000699166938139732972910, 2.49834843395888450926418408958, 3.13886182074275266895507248683, 4.23385718029115545534440390021, 5.36152747807929487804740078430, 6.26244383720660826143922967022, 7.18789805844064499005130441996, 7.74821909700373921966170481751, 8.881438228299002308126139120317, 9.618148603083933757870901966232