| L(s) = 1 | − 1.58e14·2-s + 9.80e21·3-s − 1.45e28·4-s − 7.31e32·5-s − 1.55e36·6-s − 1.37e39·7-s + 8.57e42·8-s − 2.02e45·9-s + 1.15e47·10-s − 1.77e49·11-s − 1.42e50·12-s − 4.53e52·13-s + 2.17e53·14-s − 7.17e54·15-s − 7.82e56·16-s − 3.88e57·17-s + 3.20e59·18-s − 8.75e59·19-s + 1.06e61·20-s − 1.34e61·21-s + 2.81e63·22-s − 1.90e64·23-s + 8.40e64·24-s − 1.98e66·25-s + 7.17e66·26-s − 4.06e67·27-s + 1.99e67·28-s + ⋯ |
| L(s) = 1 | − 0.795·2-s + 0.212·3-s − 0.366·4-s − 0.460·5-s − 0.169·6-s − 0.0987·7-s + 1.08·8-s − 0.954·9-s + 0.366·10-s − 0.607·11-s − 0.0780·12-s − 0.554·13-s + 0.0786·14-s − 0.0980·15-s − 0.498·16-s − 0.138·17-s + 0.759·18-s − 0.159·19-s + 0.168·20-s − 0.0210·21-s + 0.483·22-s − 0.395·23-s + 0.231·24-s − 0.787·25-s + 0.441·26-s − 0.416·27-s + 0.0362·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(96-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+95/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(48)\) |
\(\approx\) |
\(0.3664976732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3664976732\) |
| \(L(\frac{97}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.58e14T + 3.96e28T^{2} \) |
| 3 | \( 1 - 9.80e21T + 2.12e45T^{2} \) |
| 5 | \( 1 + 7.31e32T + 2.52e66T^{2} \) |
| 7 | \( 1 + 1.37e39T + 1.92e80T^{2} \) |
| 11 | \( 1 + 1.77e49T + 8.55e98T^{2} \) |
| 13 | \( 1 + 4.53e52T + 6.67e105T^{2} \) |
| 17 | \( 1 + 3.88e57T + 7.80e116T^{2} \) |
| 19 | \( 1 + 8.75e59T + 3.03e121T^{2} \) |
| 23 | \( 1 + 1.90e64T + 2.31e129T^{2} \) |
| 29 | \( 1 + 4.05e69T + 8.46e138T^{2} \) |
| 31 | \( 1 + 7.95e70T + 4.77e141T^{2} \) |
| 37 | \( 1 - 2.93e74T + 9.53e148T^{2} \) |
| 41 | \( 1 + 2.78e76T + 1.63e153T^{2} \) |
| 43 | \( 1 - 4.90e76T + 1.51e155T^{2} \) |
| 47 | \( 1 + 1.43e78T + 7.06e158T^{2} \) |
| 53 | \( 1 - 9.09e81T + 6.40e163T^{2} \) |
| 59 | \( 1 - 2.06e84T + 1.70e168T^{2} \) |
| 61 | \( 1 - 1.22e85T + 4.03e169T^{2} \) |
| 67 | \( 1 - 6.33e86T + 2.99e173T^{2} \) |
| 71 | \( 1 + 1.27e88T + 7.40e175T^{2} \) |
| 73 | \( 1 + 1.33e88T + 1.03e177T^{2} \) |
| 79 | \( 1 + 3.85e89T + 1.88e180T^{2} \) |
| 83 | \( 1 - 2.10e91T + 2.05e182T^{2} \) |
| 89 | \( 1 + 5.00e92T + 1.55e185T^{2} \) |
| 97 | \( 1 - 1.70e94T + 5.53e188T^{2} \) |
| 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
−14.68238302664352435745207324481, −13.16608894953818090014704854735, −11.32251400228692021757126190023, −9.804110537991392186863316373977, −8.528694046416708865342869143502, −7.50000988899161583434967432578, −5.39826431035492997483037586572, −3.81623778990508469786985003955, −2.15346389113113468143481976285, −0.35341403051051151834852279376,
0.35341403051051151834852279376, 2.15346389113113468143481976285, 3.81623778990508469786985003955, 5.39826431035492997483037586572, 7.50000988899161583434967432578, 8.528694046416708865342869143502, 9.804110537991392186863316373977, 11.32251400228692021757126190023, 13.16608894953818090014704854735, 14.68238302664352435745207324481
