| L(s) = 1 | − 4.61·2-s + 13.3·4-s + 3.21·5-s + 15.9·7-s − 24.5·8-s − 14.8·10-s − 54.2·11-s + 85.2·13-s − 73.8·14-s + 6.85·16-s + 48.1·17-s + 64.2·19-s + 42.8·20-s + 250.·22-s + 191.·23-s − 114.·25-s − 393.·26-s + 212.·28-s + 15.0·29-s − 209.·31-s + 164.·32-s − 222.·34-s + 51.4·35-s + 418.·37-s − 296.·38-s − 79.0·40-s − 226.·41-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.66·4-s + 0.287·5-s + 0.863·7-s − 1.08·8-s − 0.469·10-s − 1.48·11-s + 1.81·13-s − 1.40·14-s + 0.107·16-s + 0.686·17-s + 0.775·19-s + 0.479·20-s + 2.42·22-s + 1.73·23-s − 0.917·25-s − 2.97·26-s + 1.43·28-s + 0.0960·29-s − 1.21·31-s + 0.910·32-s − 1.12·34-s + 0.248·35-s + 1.85·37-s − 1.26·38-s − 0.312·40-s − 0.863·41-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
| \(L(2)\) |
\(\approx\) |
\(1.061242398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061242398\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.61T + 8T^{2} \) |
| 5 | \( 1 - 3.21T + 125T^{2} \) |
| 7 | \( 1 - 15.9T + 343T^{2} \) |
| 11 | \( 1 + 54.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 85.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 15.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 226.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 207.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 330.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 449.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 393.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 67.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 589.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 229.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 563.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.35257493500123453417370838000, −9.505534522650509995855952454012, −8.582516809960681007553560483872, −7.981987586684244619200512497706, −7.27985191461709428046410081428, −5.98998383504825999262282771259, −5.00546147515117082794995602075, −3.21418298315147243350183528985, −1.82376386299397222505541264086, −0.848784046162530097360052944030,
0.848784046162530097360052944030, 1.82376386299397222505541264086, 3.21418298315147243350183528985, 5.00546147515117082794995602075, 5.98998383504825999262282771259, 7.27985191461709428046410081428, 7.981987586684244619200512497706, 8.582516809960681007553560483872, 9.505534522650509995855952454012, 10.35257493500123453417370838000
