| L(s) = 1 | − 3.57·2-s − 9.05·3-s + 4.74·4-s + 15.7·5-s + 32.3·6-s + 11.6·8-s + 54.9·9-s − 56.1·10-s + 33.4·11-s − 42.9·12-s − 58.2·13-s − 142.·15-s − 79.4·16-s − 5.45·17-s − 196.·18-s + 30.1·19-s + 74.6·20-s − 119.·22-s − 121.·23-s − 105.·24-s + 122.·25-s + 207.·26-s − 252.·27-s − 169.·29-s + 508.·30-s − 98.7·31-s + 190.·32-s + ⋯ |
| L(s) = 1 | − 1.26·2-s − 1.74·3-s + 0.593·4-s + 1.40·5-s + 2.19·6-s + 0.513·8-s + 2.03·9-s − 1.77·10-s + 0.916·11-s − 1.03·12-s − 1.24·13-s − 2.45·15-s − 1.24·16-s − 0.0778·17-s − 2.56·18-s + 0.364·19-s + 0.835·20-s − 1.15·22-s − 1.09·23-s − 0.894·24-s + 0.980·25-s + 1.56·26-s − 1.80·27-s − 1.08·29-s + 3.09·30-s − 0.572·31-s + 1.05·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4210286627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4210286627\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 3.57T + 8T^{2} \) |
| 3 | \( 1 + 9.05T + 27T^{2} \) |
| 5 | \( 1 - 15.7T + 125T^{2} \) |
| 11 | \( 1 - 33.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.45T + 4.91e3T^{2} \) |
| 19 | \( 1 - 30.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 98.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 374.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 8.20T + 6.89e4T^{2} \) |
| 43 | \( 1 + 415.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 89.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 12.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 427.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 342.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 286.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 57.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 397.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 539.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 624.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 147.T + 9.12e5T^{2} \) |
| 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
−8.926225193793405787016456976998, −7.76965753340554676554432241720, −6.84725508924222638961328649905, −6.54066722937189681099522642792, −5.42803025929822921288118557346, −5.13720934017145254575466465110, −3.96887854017607769356012915777, −2.03501626431646707263493997088, −1.52501758538095471305352088048, −0.40025945036928002977809428379,
0.40025945036928002977809428379, 1.52501758538095471305352088048, 2.03501626431646707263493997088, 3.96887854017607769356012915777, 5.13720934017145254575466465110, 5.42803025929822921288118557346, 6.54066722937189681099522642792, 6.84725508924222638961328649905, 7.76965753340554676554432241720, 8.926225193793405787016456976998
