| L(s) = 1 | + 0.456·2-s + 3-s − 1.79·4-s + 0.456·6-s + 1.73·7-s − 1.73·8-s − 2·9-s − 2.64·11-s − 1.79·12-s + 0.791·14-s + 2.79·16-s + 4.58·17-s − 0.913·18-s + 1.73·19-s + 1.73·21-s − 1.20·22-s − 4.58·23-s − 1.73·24-s − 5·27-s − 3.10·28-s + 4.58·29-s + 6.20·31-s + 4.73·32-s − 2.64·33-s + 2.09·34-s + 3.58·36-s − 7.93·37-s + ⋯ |
| L(s) = 1 | + 0.323·2-s + 0.577·3-s − 0.895·4-s + 0.186·6-s + 0.654·7-s − 0.612·8-s − 0.666·9-s − 0.797·11-s − 0.517·12-s + 0.211·14-s + 0.697·16-s + 1.11·17-s − 0.215·18-s + 0.397·19-s + 0.377·21-s − 0.257·22-s − 0.955·23-s − 0.353·24-s − 0.962·27-s − 0.586·28-s + 0.850·29-s + 1.11·31-s + 0.837·32-s − 0.460·33-s + 0.359·34-s + 0.597·36-s − 1.30·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.456T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 17 | \( 1 - 4.58T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 + 7.93T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 1.82T + 47T^{2} \) |
| 53 | \( 1 - 7.58T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 1.00T + 67T^{2} \) |
| 71 | \( 1 - 7.02T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 6.01T + 83T^{2} \) |
| 89 | \( 1 + 9.57T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.274638125794853692190135687111, −7.63208998085026499163068394618, −6.45691459218327023560514028688, −5.48367463724769339347063586333, −5.15543189597550309596406538665, −4.23733424130308533407960939849, −3.33228885649336475196328454733, −2.74162889031913920510778341787, −1.45166116703540355379810584170, 0,
1.45166116703540355379810584170, 2.74162889031913920510778341787, 3.33228885649336475196328454733, 4.23733424130308533407960939849, 5.15543189597550309596406538665, 5.48367463724769339347063586333, 6.45691459218327023560514028688, 7.63208998085026499163068394618, 8.274638125794853692190135687111
