| L(s) = 1 | + 3-s − 7-s + 9-s − 2·11-s + 4.47·13-s − 6.47·17-s + 2·19-s − 21-s − 4·23-s + 27-s − 8.47·29-s − 0.472·31-s − 2·33-s − 2.47·37-s + 4.47·39-s − 3.52·41-s + 2.47·43-s − 6.47·47-s + 49-s − 6.47·51-s − 2·53-s + 2·57-s − 3.52·61-s − 63-s − 1.52·67-s − 4·69-s + 12.4·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 0.333·9-s − 0.603·11-s + 1.24·13-s − 1.56·17-s + 0.458·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s − 1.57·29-s − 0.0847·31-s − 0.348·33-s − 0.406·37-s + 0.716·39-s − 0.550·41-s + 0.376·43-s − 0.944·47-s + 0.142·49-s − 0.906·51-s − 0.274·53-s + 0.264·57-s − 0.451·61-s − 0.125·63-s − 0.186·67-s − 0.481·69-s + 1.48·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−8.142602719090574167477286734427, −7.37174979309425825710423080861, −6.59326432327676294972712757421, −5.92173415471408485813214942890, −5.02322496286303741528382040670, −4.03199310661157672004602699429, −3.46009477911784801728057779914, −2.45405270193974583058545922645, −1.59146660648659462035110698696, 0,
1.59146660648659462035110698696, 2.45405270193974583058545922645, 3.46009477911784801728057779914, 4.03199310661157672004602699429, 5.02322496286303741528382040670, 5.92173415471408485813214942890, 6.59326432327676294972712757421, 7.37174979309425825710423080861, 8.142602719090574167477286734427
