| L(s) = 1 | + 0.879·5-s + 4.41·7-s + 1.04·11-s + 0.0418·13-s − 4.75·17-s − 19-s + 6.87·23-s − 4.22·25-s − 3.87·29-s − 7.78·31-s + 3.87·35-s + 1.22·37-s + 12.1·41-s − 2.55·43-s + 10.4·47-s + 12.4·49-s − 2.83·53-s + 0.916·55-s + 6.16·59-s − 5.95·61-s + 0.0368·65-s + 2.59·67-s + 6.71·71-s + 8.19·73-s + 4.59·77-s + 2.40·79-s + 12.1·83-s + ⋯ |
| L(s) = 1 | + 0.393·5-s + 1.66·7-s + 0.314·11-s + 0.0116·13-s − 1.15·17-s − 0.229·19-s + 1.43·23-s − 0.845·25-s − 0.720·29-s − 1.39·31-s + 0.655·35-s + 0.201·37-s + 1.89·41-s − 0.389·43-s + 1.52·47-s + 1.78·49-s − 0.389·53-s + 0.123·55-s + 0.802·59-s − 0.762·61-s + 0.00456·65-s + 0.317·67-s + 0.797·71-s + 0.958·73-s + 0.523·77-s + 0.270·79-s + 1.33·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.839853360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.839853360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.879T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 - 0.0418T + 13T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 23 | \( 1 - 6.87T + 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 2.55T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 2.83T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 + 5.95T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 - 6.71T + 71T^{2} \) |
| 73 | \( 1 - 8.19T + 73T^{2} \) |
| 79 | \( 1 - 2.40T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 8.80T + 89T^{2} \) |
| 97 | \( 1 + 1.14T + 97T^{2} \) |
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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
−7.66545758647538496312695473612, −7.33363865647946697278288060263, −6.39976119242370983072569254778, −5.64468435328515167554647000950, −5.01448738264383409225664232801, −4.37232810206307508722963197066, −3.64834756524080992126961018295, −2.32851647169831470717609204502, −1.90278513762632051257924284710, −0.855340391729927712462145044065,
0.855340391729927712462145044065, 1.90278513762632051257924284710, 2.32851647169831470717609204502, 3.64834756524080992126961018295, 4.37232810206307508722963197066, 5.01448738264383409225664232801, 5.64468435328515167554647000950, 6.39976119242370983072569254778, 7.33363865647946697278288060263, 7.66545758647538496312695473612
