| L(s) = 1 | + 1.73·3-s + 3.46i·5-s + 10i·7-s + 2.99·9-s − 13.8·11-s + 6.92i·13-s + 5.99i·15-s + 30·17-s − 6.92·19-s + 17.3i·21-s + 12i·23-s + 13.0·25-s + 5.19·27-s − 51.9i·29-s + 14i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.692i·5-s + 1.42i·7-s + 0.333·9-s − 1.25·11-s + 0.532i·13-s + 0.399i·15-s + 1.76·17-s − 0.364·19-s + 0.824i·21-s + 0.521i·23-s + 0.520·25-s + 0.192·27-s − 1.79i·29-s + 0.451i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.33214 + 1.02219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33214 + 1.02219i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| good | 5 | \( 1 - 3.46iT - 25T^{2} \) |
| 7 | \( 1 - 10iT - 49T^{2} \) |
| 11 | \( 1 + 13.8T + 121T^{2} \) |
| 13 | \( 1 - 6.92iT - 169T^{2} \) |
| 17 | \( 1 - 30T + 289T^{2} \) |
| 19 | \( 1 + 6.92T + 361T^{2} \) |
| 23 | \( 1 - 12iT - 529T^{2} \) |
| 29 | \( 1 + 51.9iT - 841T^{2} \) |
| 31 | \( 1 - 14iT - 961T^{2} \) |
| 37 | \( 1 - 55.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 84iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 17.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 62.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 96.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 48.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 60iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 86T + 5.32e3T^{2} \) |
| 79 | \( 1 - 38iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 13.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 78T + 7.92e3T^{2} \) |
| 97 | \( 1 - 62T + 9.40e3T^{2} \) |
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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
−12.45559601933552200545983805277, −11.66088706799800358187251231675, −10.39528247997255076170063131760, −9.565289609590267713012649218809, −8.407430554934934284586356658052, −7.60185989927986674622259519328, −6.19383479836735392303422776623, −5.11556342355101203478663847995, −3.25744139473472690976504073574, −2.27893518754822178616294182480,
0.976269456267119190906010153195, 3.05837684485693998295933239549, 4.38043473603470711269553400765, 5.56229433727453279981163347647, 7.35162143486149774821864542554, 7.87780183253434360461134600129, 9.052277183254528160381941260000, 10.30019234435744986613779070923, 10.73783610113561554338586640909, 12.58829115107587574488770397325
