| L(s) = 1 | + (1.90 − 4.62i)5-s + 0.764i·7-s + 8.06i·11-s − 2.16i·13-s + 24.9·17-s + 20.7·19-s − 16.8·23-s + (−17.7 − 17.6i)25-s + 11.7i·29-s + 0.669·31-s + (3.53 + 1.45i)35-s + 48.2i·37-s + 62.5i·41-s − 71.3i·43-s − 20.2·47-s + ⋯ |
| L(s) = 1 | + (0.381 − 0.924i)5-s + 0.109i·7-s + 0.733i·11-s − 0.166i·13-s + 1.46·17-s + 1.09·19-s − 0.732·23-s + (−0.708 − 0.705i)25-s + 0.404i·29-s + 0.0215·31-s + (0.100 + 0.0417i)35-s + 1.30i·37-s + 1.52i·41-s − 1.65i·43-s − 0.431·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.307846854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.307846854\) |
| \(L(2)\) |
|
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 | \( 1 + (-1.90 + 4.62i)T \) |
| good | 7 | \( 1 - 0.764iT - 49T^{2} \) |
| 11 | \( 1 - 8.06iT - 121T^{2} \) |
| 13 | \( 1 + 2.16iT - 169T^{2} \) |
| 17 | \( 1 - 24.9T + 289T^{2} \) |
| 19 | \( 1 - 20.7T + 361T^{2} \) |
| 23 | \( 1 + 16.8T + 529T^{2} \) |
| 29 | \( 1 - 11.7iT - 841T^{2} \) |
| 31 | \( 1 - 0.669T + 961T^{2} \) |
| 37 | \( 1 - 48.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 62.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 71.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 20.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 82.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 5.39iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 83.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 10.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 87.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 15.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 33.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 55.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 22.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 68.9iT - 9.40e3T^{2} \) |
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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
−9.261782965638257982467569003572, −8.317794766001810981588208421288, −7.71506293936848182006124982809, −6.77036911920001472443321594579, −5.63460061483707352372016765204, −5.20685563744890501046241971337, −4.19768340039831293343057847597, −3.12393756303706762524778104103, −1.84644285170327531434197301998, −0.861790747569672121948473005438,
0.900847578244758673804397011780, 2.27256480922285410313957923233, 3.25723080201718811573931973225, 3.97958817412525598209481373421, 5.54013062435493109350220367953, 5.79775081418948504143443931270, 6.96027334135515651596121192874, 7.55574701152898401872602420727, 8.391074009528974418675596228831, 9.455673866011645755845564700590
