L(s) = 1 | − 15.5i·3-s − 107. i·7-s − 243·9-s − 1.20e3·13-s − 2.80e3i·19-s − 1.67e3·21-s + 3.12e3·25-s + 3.78e3i·27-s + 2.81e3i·31-s + 1.65e4·37-s + 1.87e4i·39-s − 2.40e4i·43-s + 5.27e3·49-s − 4.36e4·57-s − 3.86e4·61-s + ⋯ |
L(s) = 1 | − 0.999i·3-s − 0.828i·7-s − 9-s − 1.97·13-s − 1.78i·19-s − 0.828·21-s + 25-s + 1.00i·27-s + 0.526i·31-s + 1.98·37-s + 1.97i·39-s − 1.98i·43-s + 0.313·49-s − 1.78·57-s − 1.32·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.273384 - 1.02028i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.273384 - 1.02028i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 15.5iT \) |
good | 5 | \( 1 - 3.12e3T^{2} \) |
| 7 | \( 1 + 107. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.20e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.80e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.81e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.65e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.15e8T^{2} \) |
| 43 | \( 1 + 2.40e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.43e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.45e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.68e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.34e5T + 8.58e9T^{2} \) |
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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
−14.03699098206157437608019050711, −13.02435106977880487577777503141, −12.00894012259824749626309218686, −10.73294575317049869647731396624, −9.202446376916645672987468336133, −7.58625326679362375573330017890, −6.81550699366384216420948206783, −4.90379891855357016209382653030, −2.58921108622822011621847214912, −0.52985752654061047898380275980,
2.70826022295042964647823495122, 4.57919247105348997594189761510, 5.86379323120899748946715646826, 7.891029212429343711132222885741, 9.325957207235366741403073429107, 10.16702957579116635747158192743, 11.62697941999381711530319783869, 12.60489670257376498889653323020, 14.54987820751536153950059612806, 14.88833208845833776686730933753