| L(s) = 1 | + 1.67·2-s − 5.20·4-s + 6.76·5-s − 19.0·7-s − 22.0·8-s + 11.3·10-s + 65.4·11-s − 46.8·13-s − 31.8·14-s + 4.65·16-s − 48.0·17-s + 147.·19-s − 35.1·20-s + 109.·22-s − 33.1·23-s − 79.2·25-s − 78.4·26-s + 99.0·28-s + 73.6·29-s + 142.·31-s + 184.·32-s − 80.3·34-s − 128.·35-s + 397.·37-s + 246.·38-s − 149.·40-s + 100.·41-s + ⋯ |
| L(s) = 1 | + 0.591·2-s − 0.650·4-s + 0.604·5-s − 1.02·7-s − 0.976·8-s + 0.357·10-s + 1.79·11-s − 1.00·13-s − 0.608·14-s + 0.0727·16-s − 0.685·17-s + 1.77·19-s − 0.393·20-s + 1.06·22-s − 0.300·23-s − 0.634·25-s − 0.591·26-s + 0.668·28-s + 0.471·29-s + 0.825·31-s + 1.01·32-s − 0.405·34-s − 0.622·35-s + 1.76·37-s + 1.05·38-s − 0.590·40-s + 0.383·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.150819541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150819541\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 1.67T + 8T^{2} \) |
| 5 | \( 1 - 6.76T + 125T^{2} \) |
| 7 | \( 1 + 19.0T + 343T^{2} \) |
| 11 | \( 1 - 65.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 33.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 73.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 142.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 397.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 100.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 439.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 602.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 154.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 552.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 107.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 989.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 730.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 268.T + 9.12e5T^{2} \) |
| 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
−10.11811465325516696879513065527, −9.315855696728170898915829820670, −9.231415810337668905458344786594, −7.60735525799027211874402465432, −6.43914459523953649937623979140, −5.88375424629756848082864921235, −4.65650460576071578368958439411, −3.76399456335873750030427595255, −2.65612977154581900770117854470, −0.839625452778731815183279660024,
0.839625452778731815183279660024, 2.65612977154581900770117854470, 3.76399456335873750030427595255, 4.65650460576071578368958439411, 5.88375424629756848082864921235, 6.43914459523953649937623979140, 7.60735525799027211874402465432, 9.231415810337668905458344786594, 9.315855696728170898915829820670, 10.11811465325516696879513065527
