L(s) = 1 | + (−2.00 − 3.47i)5-s − 1.92i·7-s − 1.38i·11-s + (0.470 + 0.271i)13-s + (−1.83 − 3.17i)17-s + (1.80 − 3.96i)19-s + (−7.04 − 4.06i)23-s + (−5.53 + 9.59i)25-s + (2.40 + 1.38i)29-s − 7.06·31-s + (−6.70 + 3.86i)35-s − 2.12i·37-s + (8.24 − 4.75i)41-s + (5.88 − 3.39i)43-s + (6.18 + 3.57i)47-s + ⋯ |
L(s) = 1 | + (−0.896 − 1.55i)5-s − 0.729i·7-s − 0.417i·11-s + (0.130 + 0.0753i)13-s + (−0.444 − 0.770i)17-s + (0.413 − 0.910i)19-s + (−1.46 − 0.847i)23-s + (−1.10 + 1.91i)25-s + (0.445 + 0.257i)29-s − 1.26·31-s + (−1.13 + 0.653i)35-s − 0.349i·37-s + (1.28 − 0.743i)41-s + (0.897 − 0.518i)43-s + (0.902 + 0.520i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7613478720\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7613478720\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.80 + 3.96i)T \) |
good | 5 | \( 1 + (2.00 + 3.47i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 1.92iT - 7T^{2} \) |
| 11 | \( 1 + 1.38iT - 11T^{2} \) |
| 13 | \( 1 + (-0.470 - 0.271i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.83 + 3.17i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (7.04 + 4.06i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.40 - 1.38i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.06T + 31T^{2} \) |
| 37 | \( 1 + 2.12iT - 37T^{2} \) |
| 41 | \( 1 + (-8.24 + 4.75i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.88 + 3.39i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.18 - 3.57i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.171 + 0.0988i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.80 - 4.86i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.871 + 1.51i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.34 - 12.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.17 + 3.76i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.50 - 6.07i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.54 + 7.87i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.7iT - 83T^{2} \) |
| 89 | \( 1 + (15.3 + 8.86i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.97 + 4.60i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.576619820412081161013688940715, −7.53648291468621137381048335738, −7.21182977878180459737685021485, −5.91052881699402812173066512280, −5.16163716408130982375777817079, −4.20841506543707369012317559995, −3.98199230169911727828115396320, −2.53282168173581213352506733978, −1.06194013664209475663251511265, −0.28216678945245805570632416346,
1.87702059563315465753239426091, 2.76967888627357160950112082468, 3.69035901024804603688076542773, 4.25187535474484784068884707213, 5.71391970162411269651439786114, 6.15384636209041449892200037236, 7.09683338888304467554410907606, 7.74390017974704823998911551521, 8.256940230478718447797225191534, 9.347075331250840832156961567256