| L(s) = 1 | − 9i·3-s − 94i·7-s − 81·9-s − 146i·13-s + 46·19-s − 846·21-s + 729i·27-s + 194·31-s − 2.06e3i·37-s − 1.31e3·39-s + 3.21e3i·43-s − 6.43e3·49-s − 414i·57-s − 1.96e3·61-s + 7.61e3i·63-s + ⋯ |
| L(s) = 1 | − i·3-s − 1.91i·7-s − 9-s − 0.863i·13-s + 0.127·19-s − 1.91·21-s + 0.999i·27-s + 0.201·31-s − 1.50i·37-s − 0.863·39-s + 1.73i·43-s − 2.68·49-s − 0.127i·57-s − 0.528·61-s + 1.91i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.132708652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.132708652\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 94iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 1.46e4T^{2} \) |
| 13 | \( 1 + 146iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 8.35e4T^{2} \) |
| 19 | \( 1 - 46T + 1.30e5T^{2} \) |
| 23 | \( 1 + 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 - 194T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.06e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.21e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 4.87e6T^{2} \) |
| 53 | \( 1 + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.96e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.90e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.54e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 7.68e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.88e4iT - 8.85e7T^{2} \) |
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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
−10.70195456172704281703235907155, −9.751157445268544823063159032697, −8.318378357536452688311592724548, −7.52511937235404200572035669819, −6.88614149310971935584661024164, −5.71053272850808964196648925890, −4.26867895857641914014317401282, −3.02153227872595687725641750912, −1.33083883859862065812651018751, −0.35823566525021622926176344114,
2.12849548509117009807092110044, 3.24853304718201481237950096378, 4.67019824087582458927950128806, 5.54380542210282768278335034907, 6.47453403141592055500578710885, 8.190970242510710549382918987095, 8.990691709066853607017936745048, 9.548279674062075663627545709271, 10.68463760908044173563462002749, 11.80186086171773889268757221310
