| L(s) = 1 | + 2.82i·2-s − 8.00·4-s + 3.32i·5-s − 22.6i·8-s − 9.41·10-s − 114. i·11-s − 74.9·13-s + 64.0·16-s + 339. i·17-s − 83.3·19-s − 26.6i·20-s + 322.·22-s − 685. i·23-s + 613.·25-s − 211. i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.133i·5-s − 0.353i·8-s − 0.0941·10-s − 0.943i·11-s − 0.443·13-s + 0.250·16-s + 1.17i·17-s − 0.230·19-s − 0.0665i·20-s + 0.666·22-s − 1.29i·23-s + 0.982·25-s − 0.313i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.130202727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.130202727\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.32iT - 625T^{2} \) |
| 11 | \( 1 + 114. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 74.9T + 2.85e4T^{2} \) |
| 17 | \( 1 - 339. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 83.3T + 1.30e5T^{2} \) |
| 23 | \( 1 + 685. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 120. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 545.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 434.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 700. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 941.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 846. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 221. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 5.60e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 6.53e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 216.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.43e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.55e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.14e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 2.46e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.58e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.56e4T + 8.85e7T^{2} \) |
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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
−9.884747980143346647397826545167, −8.691706296169133912914779460591, −8.385299246884154665321596241539, −7.28079373448741755987956638632, −6.44064727774483697361553891513, −5.73571829076767211515930098431, −4.68691317082859146152073645470, −3.72075107952421561131680259639, −2.54244231000698831662816658739, −0.981153627433702129709948396962,
0.29185885125697726072869887091, 1.58849870156152123070853479151, 2.59451707367834560059413640065, 3.66827877636530377762938067680, 4.77620405399186524621303411599, 5.37714588936409782614776114913, 6.82501385968950738782697140404, 7.50618796512755715230799195264, 8.577103968809514737275862664365, 9.512370169942590967685162700091
