L(s) = 1 | + 4.68i·7-s + 2.29i·11-s + 4.97·13-s − 2.97i·17-s + 2.68i·19-s + 2.68i·23-s + 2i·29-s + 6.97·31-s + 4.39·37-s + 11.3·41-s + 9.37·43-s − 7.27i·47-s − 14.9·49-s − 2·53-s − 1.70i·59-s + ⋯ |
L(s) = 1 | + 1.77i·7-s + 0.691i·11-s + 1.38·13-s − 0.722i·17-s + 0.616i·19-s + 0.560i·23-s + 0.371i·29-s + 1.25·31-s + 0.722·37-s + 1.77·41-s + 1.42·43-s − 1.06i·47-s − 2.13·49-s − 0.274·53-s − 0.222i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0310 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0310 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.303099717\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.303099717\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.68iT - 7T^{2} \) |
| 11 | \( 1 - 2.29iT - 11T^{2} \) |
| 13 | \( 1 - 4.97T + 13T^{2} \) |
| 17 | \( 1 + 2.97iT - 17T^{2} \) |
| 19 | \( 1 - 2.68iT - 19T^{2} \) |
| 23 | \( 1 - 2.68iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 - 4.39T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 + 7.27iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 1.70iT - 59T^{2} \) |
| 61 | \( 1 - 4.58iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 0.585T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 1.02T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 - 3.95iT - 97T^{2} \) |
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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
−8.102159970553024156377027581888, −7.54572130364064354217479583025, −6.47872721233347218113411495001, −5.99429788111690908525517703179, −5.40690174290992164781560315158, −4.60828738986757344110753165831, −3.71334941716138969313287184318, −2.78477607201820678048999852324, −2.15795146686146262279957170182, −1.10242192830495995413111636945,
0.70353280188230069593065460679, 1.19001595821921610279136939980, 2.60498929409042344522251271358, 3.51735080117891841472162698646, 4.17271684919705743138056530015, 4.61443495176173417040904493399, 5.98903259175444123782006233799, 6.22497328105223757388551993484, 7.07586605133440080255716221003, 7.83095070826109574375811674568