| L(s) = 1 | + 3.37i·3-s − i·7-s − 8.37·9-s − 0.627·11-s − 1.37i·13-s − 5.37i·17-s + 6.74·19-s + 3.37·21-s − 6.74i·23-s − 18.1i·27-s − 1.37·29-s + 8·31-s − 2.11i·33-s + 2i·37-s + 4.62·39-s + ⋯ |
| L(s) = 1 | + 1.94i·3-s − 0.377i·7-s − 2.79·9-s − 0.189·11-s − 0.380i·13-s − 1.30i·17-s + 1.54·19-s + 0.735·21-s − 1.40i·23-s − 3.48i·27-s − 0.254·29-s + 1.43·31-s − 0.368i·33-s + 0.328i·37-s + 0.741·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.469468307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469468307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - 3.37iT - 3T^{2} \) |
| 11 | \( 1 + 0.627T + 11T^{2} \) |
| 13 | \( 1 + 1.37iT - 13T^{2} \) |
| 17 | \( 1 + 5.37iT - 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 + 6.74iT - 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 4.74T + 41T^{2} \) |
| 43 | \( 1 + 2.74iT - 43T^{2} \) |
| 47 | \( 1 - 10.1iT - 47T^{2} \) |
| 53 | \( 1 + 0.744iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 2.11T + 79T^{2} \) |
| 83 | \( 1 + 13.4iT - 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 + 18.8iT - 97T^{2} \) |
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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
−9.016442204175214858183311567124, −8.346182286644116696635865729859, −7.48686951755797524749638068108, −6.39021561938613912528344710055, −5.42997826449242752671567670548, −4.87017041580825592985359575967, −4.25378979804404486052084768788, −3.20554071939951152906990224633, −2.74391258124255383160731407580, −0.55373281022801603133583532460,
1.05176042708463144585098909372, 1.83827938349319940443775460060, 2.76675840188262147667750655324, 3.69845581074718933780520482144, 5.33341908392697770786119814894, 5.72779149384743419273173714540, 6.66311795570343207726577494380, 7.15810511003833892569925320170, 8.008912575360204351785984364097, 8.381276926903409748685060126835
