| L(s) = 1 | + 3.22i·3-s − 0.725i·5-s − 7-s − 7.38·9-s − 2.21i·11-s − 1.75i·13-s + 2.33·15-s + 1.69·17-s + 1.80i·19-s − 3.22i·21-s + 8.47·23-s + 4.47·25-s − 14.1i·27-s − 0.704i·29-s + 6.28·31-s + ⋯ |
| L(s) = 1 | + 1.86i·3-s − 0.324i·5-s − 0.377·7-s − 2.46·9-s − 0.668i·11-s − 0.485i·13-s + 0.603·15-s + 0.410·17-s + 0.413i·19-s − 0.703i·21-s + 1.76·23-s + 0.894·25-s − 2.72i·27-s − 0.130i·29-s + 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.681563336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.681563336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3.22iT - 3T^{2} \) |
| 5 | \( 1 + 0.725iT - 5T^{2} \) |
| 11 | \( 1 + 2.21iT - 11T^{2} \) |
| 13 | \( 1 + 1.75iT - 13T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 19 | \( 1 - 1.80iT - 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 + 0.704iT - 29T^{2} \) |
| 31 | \( 1 - 6.28T + 31T^{2} \) |
| 37 | \( 1 + 6.75iT - 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 - 7.33iT - 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 13.2iT - 53T^{2} \) |
| 59 | \( 1 + 8.86iT - 59T^{2} \) |
| 61 | \( 1 - 10.4iT - 61T^{2} \) |
| 67 | \( 1 + 1.71iT - 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 + 8.52T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 10.8iT - 83T^{2} \) |
| 89 | \( 1 - 2.13T + 89T^{2} \) |
| 97 | \( 1 + 5.73T + 97T^{2} \) |
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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
−8.898575916778454226505770732788, −8.383829913766187001688817000355, −7.35177350045029889149970513719, −6.17416711559358630480593185454, −5.55986478490288429574278020062, −4.86718942337236396560224464918, −4.19998155716978325002943220434, −3.22024197682874970223909064581, −2.86693075560200377284156254585, −0.828186775985986586670954166314,
0.72114999869440407624861409547, 1.67456677342551020654240088338, 2.64528619660099577306327279744, 3.25693552302596931122338238538, 4.73168083595928322630581453161, 5.54955334758034045068012622431, 6.52896531091851137119419088447, 6.98411441577916103640533554563, 7.25434824519567620813081512791, 8.371538326342908571915163812801
