| L(s) = 1 | − 1.36i·5-s − 1.24i·7-s + 5.79·11-s − 16.3i·13-s + 5.01·17-s + 26.1·19-s + 25.1i·23-s + 23.1·25-s + 32.7i·29-s − 1.01i·31-s − 1.69·35-s + 14.9i·37-s − 72.5·41-s − 33.4·43-s − 66.5i·47-s + ⋯ |
| L(s) = 1 | − 0.272i·5-s − 0.177i·7-s + 0.527·11-s − 1.26i·13-s + 0.294·17-s + 1.37·19-s + 1.09i·23-s + 0.925·25-s + 1.13i·29-s − 0.0328i·31-s − 0.0484·35-s + 0.405i·37-s − 1.76·41-s − 0.778·43-s − 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.219840331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.219840331\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.36iT - 25T^{2} \) |
| 7 | \( 1 + 1.24iT - 49T^{2} \) |
| 11 | \( 1 - 5.79T + 121T^{2} \) |
| 13 | \( 1 + 16.3iT - 169T^{2} \) |
| 17 | \( 1 - 5.01T + 289T^{2} \) |
| 19 | \( 1 - 26.1T + 361T^{2} \) |
| 23 | \( 1 - 25.1iT - 529T^{2} \) |
| 29 | \( 1 - 32.7iT - 841T^{2} \) |
| 31 | \( 1 + 1.01iT - 961T^{2} \) |
| 37 | \( 1 - 14.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 72.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 66.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 54.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 20.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 80.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 113.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 21.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 160.T + 9.40e3T^{2} \) |
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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
−8.745448007864107505225247346569, −7.934209264561403387153941214287, −7.25588200780231331468176727280, −6.45492783898497139105200979020, −5.28567784455167277621529216870, −5.07575882645637556408809389925, −3.56973385499537453755880207284, −3.16799847486269374581430298621, −1.62537491323022026127880302280, −0.69269790404052235843934253777,
0.949242762933844009594424728187, 2.11226945474570591394578935607, 3.12464452205437913839372345378, 4.07974353103406177357018521835, 4.89911733468991204155770309187, 5.85299156132553213815318284715, 6.72973412072127079651682045325, 7.18313091002048635372108204743, 8.265429002067747136729751170898, 8.912153757567593812456521218752
