L(s) = 1 | − 2-s − 2.56·3-s + 4-s + 1.93·5-s + 2.56·6-s − 4.25·7-s − 8-s + 3.58·9-s − 1.93·10-s − 1.60·11-s − 2.56·12-s + 0.921·13-s + 4.25·14-s − 4.97·15-s + 16-s + 7.21·17-s − 3.58·18-s − 2.67·19-s + 1.93·20-s + 10.9·21-s + 1.60·22-s + 0.490·23-s + 2.56·24-s − 1.24·25-s − 0.921·26-s − 1.48·27-s − 4.25·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.48·3-s + 0.5·4-s + 0.866·5-s + 1.04·6-s − 1.60·7-s − 0.353·8-s + 1.19·9-s − 0.612·10-s − 0.484·11-s − 0.740·12-s + 0.255·13-s + 1.13·14-s − 1.28·15-s + 0.250·16-s + 1.75·17-s − 0.843·18-s − 0.612·19-s + 0.433·20-s + 2.38·21-s + 0.342·22-s + 0.102·23-s + 0.523·24-s − 0.249·25-s − 0.180·26-s − 0.286·27-s − 0.803·28-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
\(L(1)\) |
\(\approx\) |
\(0.4836581591\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4836581591\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 1.93T + 5T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 13 | \( 1 - 0.921T + 13T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 - 0.490T + 23T^{2} \) |
| 29 | \( 1 - 6.73T + 29T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 + 4.99T + 37T^{2} \) |
| 41 | \( 1 + 6.91T + 41T^{2} \) |
| 43 | \( 1 + 4.79T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 1.18T + 53T^{2} \) |
| 59 | \( 1 - 0.965T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 5.16T + 67T^{2} \) |
| 71 | \( 1 - 9.73T + 71T^{2} \) |
| 73 | \( 1 + 0.507T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 - 8.80T + 83T^{2} \) |
| 89 | \( 1 + 2.21T + 89T^{2} \) |
| 97 | \( 1 + 1.70T + 97T^{2} \) |
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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
−8.574912743879899027002193599893, −7.52348494109966585103274771953, −6.76420597679176148732435788151, −6.24329562546668825850929626271, −5.67222992035298676143100192956, −5.15187183699525094323070088973, −3.71493306777785627303675415629, −2.87488340061231768967452831221, −1.62048079103672751690072832560, −0.47958425241851116486091685650,
0.47958425241851116486091685650, 1.62048079103672751690072832560, 2.87488340061231768967452831221, 3.71493306777785627303675415629, 5.15187183699525094323070088973, 5.67222992035298676143100192956, 6.24329562546668825850929626271, 6.76420597679176148732435788151, 7.52348494109966585103274771953, 8.574912743879899027002193599893