| L(s) = 1 | + 3i·3-s − 3.46i·5-s + 24.2·7-s − 9·9-s − 48i·11-s + 41.5i·13-s + 10.3·15-s + 54·17-s − 4i·19-s + 72.7i·21-s + 173.·23-s + 113·25-s − 27i·27-s + 162. i·29-s + 58.8·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.309i·5-s + 1.30·7-s − 0.333·9-s − 1.31i·11-s + 0.886i·13-s + 0.178·15-s + 0.770·17-s − 0.0482i·19-s + 0.755i·21-s + 1.57·23-s + 0.904·25-s − 0.192i·27-s + 1.04i·29-s + 0.341·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.00643 + 0.264152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00643 + 0.264152i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| good | 5 | \( 1 + 3.46iT - 125T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 11 | \( 1 + 48iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 41.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 294T + 6.89e4T^{2} \) |
| 43 | \( 1 + 188iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 744. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 252iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 90.0iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 628iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 720iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−11.79074829856699289253686857069, −11.22289682789172044079495797682, −10.28790349129504435599086789473, −8.842873712942778089841656892782, −8.447021118912845150489565926778, −6.98598970870768570764569196367, −5.45267556693726555884314250413, −4.66810372483814567510359672167, −3.20261935189415211060644030745, −1.23184531978919610774676607727,
1.24878618868391497694858730949, 2.69348918937598456249345726662, 4.55380105749821572389403599080, 5.62342742433537457305644356755, 7.16634662060996807894710402171, 7.74016892317729523793644520842, 8.888572529015301387636675204298, 10.24184191282210251100291666770, 11.09520441905877589515366024542, 12.12558169297969469295506078846
