| L(s) = 1 | − 2-s − 0.469·3-s + 4-s + 0.405·5-s + 0.469·6-s + 2.00·7-s − 8-s − 2.77·9-s − 0.405·10-s − 2.46·11-s − 0.469·12-s − 5.49·13-s − 2.00·14-s − 0.190·15-s + 16-s − 3.55·17-s + 2.77·18-s + 7.59·19-s + 0.405·20-s − 0.940·21-s + 2.46·22-s − 4.38·23-s + 0.469·24-s − 4.83·25-s + 5.49·26-s + 2.71·27-s + 2.00·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.270·3-s + 0.5·4-s + 0.181·5-s + 0.191·6-s + 0.757·7-s − 0.353·8-s − 0.926·9-s − 0.128·10-s − 0.744·11-s − 0.135·12-s − 1.52·13-s − 0.535·14-s − 0.0490·15-s + 0.250·16-s − 0.861·17-s + 0.655·18-s + 1.74·19-s + 0.0906·20-s − 0.205·21-s + 0.526·22-s − 0.914·23-s + 0.0957·24-s − 0.967·25-s + 1.07·26-s + 0.521·27-s + 0.378·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8022689777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8022689777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 0.469T + 3T^{2} \) |
| 5 | \( 1 - 0.405T + 5T^{2} \) |
| 7 | \( 1 - 2.00T + 7T^{2} \) |
| 11 | \( 1 + 2.46T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 - 7.59T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 + 7.15T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 + 4.66T + 41T^{2} \) |
| 43 | \( 1 - 0.568T + 43T^{2} \) |
| 47 | \( 1 - 3.96T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 3.31T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 5.73T + 71T^{2} \) |
| 73 | \( 1 - 6.75T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 5.57T + 83T^{2} \) |
| 89 | \( 1 + 3.60T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.327937279967794959296687790055, −7.76755165877254665950162483774, −7.25310773471601143308146181953, −6.25607906009593809684097814923, −5.32328801628241995334414899345, −5.08187773947299420794235547362, −3.76382220142840799539444781781, −2.55775517092291367597859912788, −2.06854879890190153916435733673, −0.54818685516436951386257442887,
0.54818685516436951386257442887, 2.06854879890190153916435733673, 2.55775517092291367597859912788, 3.76382220142840799539444781781, 5.08187773947299420794235547362, 5.32328801628241995334414899345, 6.25607906009593809684097814923, 7.25310773471601143308146181953, 7.76755165877254665950162483774, 8.327937279967794959296687790055
