| L(s) = 1 | − 13.8i·7-s + 22i·13-s − 27.7·19-s + 25·25-s − 41.5i·31-s − 26i·37-s − 83.1·43-s − 142.·49-s + 74i·61-s + 55.4·67-s − 46·73-s + 69.2i·79-s + 304.·91-s − 2·97-s + 69.2i·103-s + ⋯ |
| L(s) = 1 | − 1.97i·7-s + 1.69i·13-s − 1.45·19-s + 25-s − 1.34i·31-s − 0.702i·37-s − 1.93·43-s − 2.91·49-s + 1.21i·61-s + 0.827·67-s − 0.630·73-s + 0.876i·79-s + 3.34·91-s − 0.0206·97-s + 0.672i·103-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1304372079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1304372079\) |
| \(L(2)\) |
|
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 | \( 1 - 25T^{2} \) |
| 7 | \( 1 + 13.8iT - 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 - 22iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + 27.7T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 41.5iT - 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + 83.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 - 74iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 55.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 69.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 + 2T + 9.40e3T^{2} \) |
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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
−9.124313139632998814998590012677, −8.320261923174406983529104869239, −7.43603731137191681785903717277, −6.79793837939358689559362310046, −6.33021216588608360028369257446, −4.86973399648162075734635453368, −4.20743777020550089659871819793, −3.69844199999444156628778317366, −2.22939913359474318135192884964, −1.18065369530714239652909557510,
0.03188909107629850205720401132, 1.68045250254813599180321532949, 2.72572708795792804742666724951, 3.28834548187421380994412350410, 4.83247014835427112347868630950, 5.32866737403374284677584309886, 6.14353094764651005355899274442, 6.80917444017758316539479661090, 8.218901166673824556344738245210, 8.375000674162922534991475856835
