| L(s) = 1 | − 0.349·2-s + 1.74·3-s − 1.87·4-s + 1.36·5-s − 0.609·6-s − 0.0430·7-s + 1.35·8-s + 0.0478·9-s − 0.476·10-s − 1.16·11-s − 3.27·12-s + 6.53·13-s + 0.0150·14-s + 2.38·15-s + 3.28·16-s + 5.57·17-s − 0.0167·18-s + 1.21·19-s − 2.56·20-s − 0.0752·21-s + 0.405·22-s + 0.469·23-s + 2.36·24-s − 3.13·25-s − 2.28·26-s − 5.15·27-s + 0.0809·28-s + ⋯ |
| L(s) = 1 | − 0.246·2-s + 1.00·3-s − 0.939·4-s + 0.610·5-s − 0.248·6-s − 0.0162·7-s + 0.478·8-s + 0.0159·9-s − 0.150·10-s − 0.349·11-s − 0.946·12-s + 1.81·13-s + 0.00402·14-s + 0.615·15-s + 0.820·16-s + 1.35·17-s − 0.00393·18-s + 0.279·19-s − 0.573·20-s − 0.0164·21-s + 0.0863·22-s + 0.0979·23-s + 0.482·24-s − 0.627·25-s − 0.447·26-s − 0.991·27-s + 0.0152·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.646414285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.646414285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 + 0.349T + 2T^{2} \) |
| 3 | \( 1 - 1.74T + 3T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 7 | \( 1 + 0.0430T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 - 6.53T + 13T^{2} \) |
| 17 | \( 1 - 5.57T + 17T^{2} \) |
| 19 | \( 1 - 1.21T + 19T^{2} \) |
| 23 | \( 1 - 0.469T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 - 3.80T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 - 4.43T + 43T^{2} \) |
| 47 | \( 1 + 9.22T + 47T^{2} \) |
| 53 | \( 1 + 4.85T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 8.90T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 9.01T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 4.36T + 97T^{2} \) |
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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
−10.60308643918429435168961792039, −9.800301190351391324784015472892, −9.106780763897687494794455854225, −8.305309017115121005217997324627, −7.80504356284910383861965643930, −6.17334947167989597003140480878, −5.31902154891760224431924628853, −3.91311793791101593463119497161, −3.03522627534085960617825045836, −1.35908993848087412859656710800,
1.35908993848087412859656710800, 3.03522627534085960617825045836, 3.91311793791101593463119497161, 5.31902154891760224431924628853, 6.17334947167989597003140480878, 7.80504356284910383861965643930, 8.305309017115121005217997324627, 9.106780763897687494794455854225, 9.800301190351391324784015472892, 10.60308643918429435168961792039
