| L(s) = 1 | − 8.89·5-s − 2.82i·7-s − 18.2i·11-s − 5.79·13-s + 21.5·17-s + 18.2i·19-s − 33.3i·23-s + 54.1·25-s + 4.49·29-s + 2.25i·31-s + 25.1i·35-s + 43.1·37-s − 1.59·41-s − 63.4i·43-s − 72.3i·47-s + ⋯ |
| L(s) = 1 | − 1.77·5-s − 0.404i·7-s − 1.65i·11-s − 0.445·13-s + 1.27·17-s + 0.960i·19-s − 1.45i·23-s + 2.16·25-s + 0.154·29-s + 0.0728i·31-s + 0.719i·35-s + 1.16·37-s − 0.0389·41-s − 1.47i·43-s − 1.54i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5841973997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5841973997\) |
| \(L(2)\) |
|
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 | \( 1 + 8.89T + 25T^{2} \) |
| 7 | \( 1 + 2.82iT - 49T^{2} \) |
| 11 | \( 1 + 18.2iT - 121T^{2} \) |
| 13 | \( 1 + 5.79T + 169T^{2} \) |
| 17 | \( 1 - 21.5T + 289T^{2} \) |
| 19 | \( 1 - 18.2iT - 361T^{2} \) |
| 23 | \( 1 + 33.3iT - 529T^{2} \) |
| 29 | \( 1 - 4.49T + 841T^{2} \) |
| 31 | \( 1 - 2.25iT - 961T^{2} \) |
| 37 | \( 1 - 43.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.59T + 1.68e3T^{2} \) |
| 43 | \( 1 + 63.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 72.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 70.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 63.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 3.24iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 68.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 35.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 42.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 5.19T + 7.92e3T^{2} \) |
| 97 | \( 1 + 26.8T + 9.40e3T^{2} \) |
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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
−8.255848613980714279835080484746, −7.83757159966214314812262812384, −7.08004765387898727351035803610, −6.12720467142908382261102874064, −5.20839565418313937306978203604, −4.14820758836851513516241260921, −3.60892003586653144188670186164, −2.82902890823686938854746640171, −0.961513046148331989969668791188, −0.19351709244857873202737003643,
1.24688137902644227225060261617, 2.68269918595148770658814110693, 3.51115450056841742785751836941, 4.53331839538180542088612902796, 4.90613893897824022175001862267, 6.17006129364625411946284798922, 7.34271036486710368631466596640, 7.52153164703328716819102538242, 8.181511134600720512482881887376, 9.385037737336558790171228585157
