L(s) = 1 | + 7.65i·5-s − 1.65i·7-s + 1.51·11-s − 0.343i·13-s + 13.3·17-s − 20.8·19-s − 33.6i·23-s − 33.6·25-s − 39.6i·29-s − 45.2i·31-s + 12.6·35-s + 29.5i·37-s + 24.6·41-s − 50.0·43-s − 35.3i·47-s + ⋯ |
L(s) = 1 | + 1.53i·5-s − 0.236i·7-s + 0.137·11-s − 0.0263i·13-s + 0.783·17-s − 1.09·19-s − 1.46i·23-s − 1.34·25-s − 1.36i·29-s − 1.45i·31-s + 0.362·35-s + 0.799i·37-s + 0.600·41-s − 1.16·43-s − 0.751i·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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.479120424\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.479120424\) |
\(L(2)\) |
|
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 | \( 1 - 7.65iT - 25T^{2} \) |
| 7 | \( 1 + 1.65iT - 49T^{2} \) |
| 11 | \( 1 - 1.51T + 121T^{2} \) |
| 13 | \( 1 + 0.343iT - 169T^{2} \) |
| 17 | \( 1 - 13.3T + 289T^{2} \) |
| 19 | \( 1 + 20.8T + 361T^{2} \) |
| 23 | \( 1 + 33.6iT - 529T^{2} \) |
| 29 | \( 1 + 39.6iT - 841T^{2} \) |
| 31 | \( 1 + 45.2iT - 961T^{2} \) |
| 37 | \( 1 - 29.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 24.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 50.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 16.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 53.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 34.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 62.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 40.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 55.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 137. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 114.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 2.56T + 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−8.595514517154037714464494202485, −7.88102477566266680857270892340, −7.12838586851016106256157466253, −6.38926891708235939272955223444, −5.91156140288041633559039271448, −4.55060255459218363844278020617, −3.78614723075297534936454281179, −2.81842109048989462762305705108, −2.08823141015504659123719961807, −0.39934870548540932679642423866,
1.03311096386296363082843117828, 1.80952398725104485962192007416, 3.24853457834024700000537156027, 4.16134051133498346549994040868, 5.10023391819224219935564446176, 5.51058199137491943019382806257, 6.55050894617484226214716211335, 7.54093927669292454497108168751, 8.283303300882393071772395280140, 8.977205352855213875495138897753