| L(s) = 1 | − 1.73i·3-s − 5.58·5-s − 2.99·9-s + 4.37i·11-s − 17.1·13-s + 9.66i·15-s − 0.747·17-s − 3.65i·19-s + 9.86i·23-s + 6.16·25-s + 5.19i·27-s − 2·29-s − 17.8i·31-s + 7.58·33-s − 7.49·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.11·5-s − 0.333·9-s + 0.397i·11-s − 1.32·13-s + 0.644i·15-s − 0.0439·17-s − 0.192i·19-s + 0.428i·23-s + 0.246·25-s + 0.192i·27-s − 0.0689·29-s − 0.577i·31-s + 0.229·33-s − 0.202·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9113623778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9113623778\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5.58T + 25T^{2} \) |
| 11 | \( 1 - 4.37iT - 121T^{2} \) |
| 13 | \( 1 + 17.1T + 169T^{2} \) |
| 17 | \( 1 + 0.747T + 289T^{2} \) |
| 19 | \( 1 + 3.65iT - 361T^{2} \) |
| 23 | \( 1 - 9.86iT - 529T^{2} \) |
| 29 | \( 1 + 2T + 841T^{2} \) |
| 31 | \( 1 + 17.8iT - 961T^{2} \) |
| 37 | \( 1 + 7.49T + 1.36e3T^{2} \) |
| 41 | \( 1 + 76.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 70.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 89.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 120.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 23.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 66.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 95.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 34.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 52.5T + 9.40e3T^{2} \) |
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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
−8.509771782903048628103898258777, −7.88180747480527625616262050560, −7.23933934262005174456543463471, −6.72670313968016320049811691564, −5.53235187102480080306792410962, −4.73421578266227239571332649285, −3.87338681561528541604838307060, −2.88021281751218388185183933981, −1.87471093353470660055538753674, −0.46071544798692331345826567640,
0.48659485308720544301425632875, 2.20544564036786792349962734169, 3.30423873242890170150474385739, 3.96537874296615215659723726840, 4.85731647006165248382884759915, 5.48986672080343787235859295435, 6.73740526917700873355361495129, 7.33057219895812236854570989579, 8.255704903875963404232718963966, 8.683962726358685461391782037943
