| L(s) = 1 | + (15.4 + 1.94i)3-s + 101.·5-s − 200. i·7-s + (235. + 60.0i)9-s + 350. i·11-s − 744. i·13-s + (1.56e3 + 196. i)15-s − 1.44e3i·17-s − 1.31e3·19-s + (389. − 3.09e3i)21-s + 2.04e3·23-s + 7.09e3·25-s + (3.52e3 + 1.38e3i)27-s − 3.78e3·29-s + 1.42e3i·31-s + ⋯ |
| L(s) = 1 | + (0.992 + 0.124i)3-s + 1.80·5-s − 1.54i·7-s + (0.968 + 0.247i)9-s + 0.872i·11-s − 1.22i·13-s + (1.79 + 0.225i)15-s − 1.21i·17-s − 0.834·19-s + (0.192 − 1.53i)21-s + 0.804·23-s + 2.27·25-s + (0.930 + 0.366i)27-s − 0.835·29-s + 0.266i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.499329854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.499329854\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-15.4 - 1.94i)T \) |
| good | 5 | \( 1 - 101.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 200. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 350. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 744. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.44e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.42e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 2.61e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.08e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.41e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 539.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.37e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 5.04e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.32e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 7.58e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.73e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−10.21691572513241641660237704242, −9.634115865571054527032009245603, −8.738126278324187260765480842764, −7.39827294172997388567917124379, −6.89306916405399800396576566582, −5.43399178607196963308608077145, −4.43614741351503784521742380539, −3.08567110481146756769231958322, −2.04744658672281582410252303239, −0.951855464419292732233359826290,
1.69132180644767741556776644232, 2.13605907112522744543531056664, 3.26592178923222946732891111527, 4.91495323795638476085416242663, 6.08498889867385067861785000228, 6.52532830920824039081246602054, 8.274728155273731844406636886147, 8.981299401992340257428777053645, 9.392609943989547782976299641906, 10.39754151164990002357351952849
