| L(s) = 1 | − 0.227·2-s − 1.80·3-s − 7.94·4-s − 5·5-s + 0.411·6-s + 3.62·8-s − 23.7·9-s + 1.13·10-s + 17.7·11-s + 14.3·12-s − 62.3·13-s + 9.04·15-s + 62.7·16-s + 87.1·17-s + 5.39·18-s − 101.·19-s + 39.7·20-s − 4.02·22-s + 93.6·23-s − 6.56·24-s + 25·25-s + 14.1·26-s + 91.7·27-s + 297.·29-s − 2.05·30-s + 91.2·31-s − 43.3·32-s + ⋯ |
| L(s) = 1 | − 0.0804·2-s − 0.348·3-s − 0.993·4-s − 0.447·5-s + 0.0279·6-s + 0.160·8-s − 0.878·9-s + 0.0359·10-s + 0.485·11-s + 0.345·12-s − 1.32·13-s + 0.155·15-s + 0.980·16-s + 1.24·17-s + 0.0706·18-s − 1.22·19-s + 0.444·20-s − 0.0390·22-s + 0.848·23-s − 0.0558·24-s + 0.200·25-s + 0.106·26-s + 0.653·27-s + 1.90·29-s − 0.0125·30-s + 0.528·31-s − 0.239·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8076596089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8076596089\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.227T + 8T^{2} \) |
| 3 | \( 1 + 1.80T + 27T^{2} \) |
| 11 | \( 1 - 17.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 93.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 297.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 91.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 281.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.81T + 7.95e4T^{2} \) |
| 47 | \( 1 + 92.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 99.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 434.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 461.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 518.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 542.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 239.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 299.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 288.T + 9.12e5T^{2} \) |
| 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
−11.89146406539261853396026217037, −10.59410912136563073605155120952, −9.731316077662657391400936019235, −8.668288080242733655854396798452, −7.922521658700982484838008990117, −6.56049657469885735673878382480, −5.27261652133732265511669121279, −4.41792883063734429864965577134, −2.97707109187121412549108112270, −0.67130409657206205598495991732,
0.67130409657206205598495991732, 2.97707109187121412549108112270, 4.41792883063734429864965577134, 5.27261652133732265511669121279, 6.56049657469885735673878382480, 7.922521658700982484838008990117, 8.668288080242733655854396798452, 9.731316077662657391400936019235, 10.59410912136563073605155120952, 11.89146406539261853396026217037
