| L(s) = 1 | − 0.921·2-s − 0.437·3-s − 1.15·4-s − 1.68·5-s + 0.403·6-s − 3.39·7-s + 2.90·8-s − 2.80·9-s + 1.54·10-s − 5.13·11-s + 0.503·12-s − 5.70·13-s + 3.12·14-s + 0.735·15-s − 0.375·16-s − 17-s + 2.58·18-s + 5.21·19-s + 1.93·20-s + 1.48·21-s + 4.73·22-s + 5.80·23-s − 1.27·24-s − 2.17·25-s + 5.26·26-s + 2.54·27-s + 3.90·28-s + ⋯ |
| L(s) = 1 | − 0.651·2-s − 0.252·3-s − 0.575·4-s − 0.751·5-s + 0.164·6-s − 1.28·7-s + 1.02·8-s − 0.936·9-s + 0.490·10-s − 1.54·11-s + 0.145·12-s − 1.58·13-s + 0.835·14-s + 0.189·15-s − 0.0938·16-s − 0.242·17-s + 0.610·18-s + 1.19·19-s + 0.432·20-s + 0.323·21-s + 1.00·22-s + 1.20·23-s − 0.259·24-s − 0.434·25-s + 1.03·26-s + 0.489·27-s + 0.737·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1366174462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1366174462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 0.921T + 2T^{2} \) |
| 3 | \( 1 + 0.437T + 3T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 19 | \( 1 - 5.21T + 19T^{2} \) |
| 23 | \( 1 - 5.80T + 23T^{2} \) |
| 29 | \( 1 + 5.98T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 + 7.87T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 + 8.30T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 61 | \( 1 - 3.96T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 - 5.35T + 71T^{2} \) |
| 73 | \( 1 - 2.06T + 73T^{2} \) |
| 79 | \( 1 + 2.79T + 79T^{2} \) |
| 83 | \( 1 - 8.38T + 83T^{2} \) |
| 89 | \( 1 - 2.17T + 89T^{2} \) |
| 97 | \( 1 - 4.81T + 97T^{2} \) |
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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
−9.877777750318962968633299188830, −9.268867624616078250184697963803, −8.285112814308865584945095710809, −7.59811181633512374606720710193, −6.89245441803338762846190111048, −5.40442310085505941611854170600, −4.97172636147764318092519174706, −3.52051433471446646431857712032, −2.65250483361022711249722468971, −0.29832937530226945046371510169,
0.29832937530226945046371510169, 2.65250483361022711249722468971, 3.52051433471446646431857712032, 4.97172636147764318092519174706, 5.40442310085505941611854170600, 6.89245441803338762846190111048, 7.59811181633512374606720710193, 8.285112814308865584945095710809, 9.268867624616078250184697963803, 9.877777750318962968633299188830
