| L(s) = 1 | + 2.04·2-s + 3·3-s − 3.82·4-s + 12.0·5-s + 6.12·6-s + 29.7·7-s − 24.1·8-s + 9·9-s + 24.6·10-s + 28.0·11-s − 11.4·12-s + 60.7·14-s + 36.2·15-s − 18.7·16-s − 50.6·17-s + 18.3·18-s + 105.·19-s − 46.2·20-s + 89.2·21-s + 57.3·22-s − 160.·23-s − 72.4·24-s + 20.9·25-s + 27·27-s − 113.·28-s + 140.·29-s + 74.0·30-s + ⋯ |
| L(s) = 1 | + 0.722·2-s + 0.577·3-s − 0.478·4-s + 1.08·5-s + 0.416·6-s + 1.60·7-s − 1.06·8-s + 0.333·9-s + 0.780·10-s + 0.769·11-s − 0.276·12-s + 1.15·14-s + 0.623·15-s − 0.292·16-s − 0.722·17-s + 0.240·18-s + 1.26·19-s − 0.517·20-s + 0.927·21-s + 0.555·22-s − 1.45·23-s − 0.616·24-s + 0.167·25-s + 0.192·27-s − 0.768·28-s + 0.897·29-s + 0.450·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.357729973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.357729973\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.04T + 8T^{2} \) |
| 5 | \( 1 - 12.0T + 125T^{2} \) |
| 7 | \( 1 - 29.7T + 343T^{2} \) |
| 11 | \( 1 - 28.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 50.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 228.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 36.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 438.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 286.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 537.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 75.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 17.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 334.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 748.T + 9.12e5T^{2} \) |
| 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
−10.34530232900344841211890218280, −9.554523591662090609105589733150, −8.703621136640195298166004335811, −8.017789492728834418360882486773, −6.63168203453898250400688686755, −5.57356616374405487280386650143, −4.78020390147140636493623910844, −3.87697140593372696098422937258, −2.42467376604739708071354213983, −1.32276851978924192492851200886,
1.32276851978924192492851200886, 2.42467376604739708071354213983, 3.87697140593372696098422937258, 4.78020390147140636493623910844, 5.57356616374405487280386650143, 6.63168203453898250400688686755, 8.017789492728834418360882486773, 8.703621136640195298166004335811, 9.554523591662090609105589733150, 10.34530232900344841211890218280
