| L(s) = 1 | − 0.386·3-s − 3.17·5-s − 3.44·7-s − 2.85·9-s − 3.44·11-s + 1.22·15-s − 1.77·17-s − 3.33·19-s + 1.33·21-s + 0.386·23-s + 5.07·25-s + 2.26·27-s − 4.57·29-s − 11.0·31-s + 1.33·33-s + 10.9·35-s + 1.62·37-s − 2.26·41-s − 10.7·43-s + 9.04·45-s − 8.11·47-s + 4.85·49-s + 0.685·51-s + 11.6·53-s + 10.9·55-s + 1.28·57-s + 6.32·59-s + ⋯ |
| L(s) = 1 | − 0.223·3-s − 1.41·5-s − 1.30·7-s − 0.950·9-s − 1.03·11-s + 0.316·15-s − 0.430·17-s − 0.764·19-s + 0.290·21-s + 0.0805·23-s + 1.01·25-s + 0.435·27-s − 0.849·29-s − 1.98·31-s + 0.231·33-s + 1.84·35-s + 0.266·37-s − 0.354·41-s − 1.63·43-s + 1.34·45-s − 1.18·47-s + 0.692·49-s + 0.0959·51-s + 1.60·53-s + 1.47·55-s + 0.170·57-s + 0.823·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08534023457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08534023457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.386T + 3T^{2} \) |
| 5 | \( 1 + 3.17T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 - 0.386T + 23T^{2} \) |
| 29 | \( 1 + 4.57T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 8.11T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 - 2.42T + 61T^{2} \) |
| 67 | \( 1 - 9.14T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 8.40T + 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 + 1.11T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 4.48T + 97T^{2} \) |
| 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
−8.667614965565705896428426054603, −8.158039274458161539329821621107, −7.27530984160419824275845599678, −6.66894779916955244575736862128, −5.72831592479153046398081590413, −4.98340819878695095957373530243, −3.81020402386030550702407536809, −3.34239416422248359006747810188, −2.32986820345926741717739082766, −0.16842961856690561817906113073,
0.16842961856690561817906113073, 2.32986820345926741717739082766, 3.34239416422248359006747810188, 3.81020402386030550702407536809, 4.98340819878695095957373530243, 5.72831592479153046398081590413, 6.66894779916955244575736862128, 7.27530984160419824275845599678, 8.158039274458161539329821621107, 8.667614965565705896428426054603
