| L(s) = 1 | + 2.29·2-s − 3·3-s − 2.73·4-s − 14.4·5-s − 6.88·6-s − 20.5·7-s − 24.6·8-s + 9·9-s − 33.0·10-s + 21.2·11-s + 8.21·12-s − 51.9·13-s − 47.0·14-s + 43.2·15-s − 34.6·16-s − 61.2·17-s + 20.6·18-s + 100.·19-s + 39.4·20-s + 61.5·21-s + 48.7·22-s + 73.9·23-s + 73.8·24-s + 82.6·25-s − 119.·26-s − 27·27-s + 56.1·28-s + ⋯ |
| L(s) = 1 | + 0.811·2-s − 0.577·3-s − 0.342·4-s − 1.28·5-s − 0.468·6-s − 1.10·7-s − 1.08·8-s + 0.333·9-s − 1.04·10-s + 0.582·11-s + 0.197·12-s − 1.10·13-s − 0.898·14-s + 0.744·15-s − 0.540·16-s − 0.874·17-s + 0.270·18-s + 1.21·19-s + 0.441·20-s + 0.639·21-s + 0.472·22-s + 0.669·23-s + 0.628·24-s + 0.661·25-s − 0.899·26-s − 0.192·27-s + 0.379·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6698613924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6698613924\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + 157T \) |
| good | 2 | \( 1 - 2.29T + 8T^{2} \) |
| 5 | \( 1 + 14.4T + 125T^{2} \) |
| 7 | \( 1 + 20.5T + 343T^{2} \) |
| 11 | \( 1 - 21.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 51.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 233.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 91.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 418.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 337.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 177.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 133.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 438.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 723.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 491.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 353.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 398.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 329.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 328.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.50e3T + 9.12e5T^{2} \) |
| 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
−10.86376795808284789931694070989, −9.586704624408666371885782685735, −9.030513644822705506757250944230, −7.58102864093715787161913131289, −6.83001372846632495641681258888, −5.75241789582087187036822620763, −4.69893670620437424458969532584, −3.90447140442425526988550652851, −2.98228098478581253569142934654, −0.45023885641466007878609120264,
0.45023885641466007878609120264, 2.98228098478581253569142934654, 3.90447140442425526988550652851, 4.69893670620437424458969532584, 5.75241789582087187036822620763, 6.83001372846632495641681258888, 7.58102864093715787161913131289, 9.030513644822705506757250944230, 9.586704624408666371885782685735, 10.86376795808284789931694070989
