| L(s) = 1 | − 1.90·2-s − 0.428·3-s + 1.63·4-s + 1.47·5-s + 0.816·6-s + 0.702·8-s − 2.81·9-s − 2.80·10-s − 4.39·11-s − 0.698·12-s − 0.631·15-s − 4.60·16-s + 1.20·17-s + 5.36·18-s − 3.24·19-s + 2.40·20-s + 8.37·22-s − 4.43·23-s − 0.301·24-s − 2.82·25-s + 2.49·27-s + 0.167·29-s + 1.20·30-s − 5.24·31-s + 7.36·32-s + 1.88·33-s − 2.29·34-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 0.247·3-s + 0.815·4-s + 0.658·5-s + 0.333·6-s + 0.248·8-s − 0.938·9-s − 0.887·10-s − 1.32·11-s − 0.201·12-s − 0.162·15-s − 1.15·16-s + 0.291·17-s + 1.26·18-s − 0.743·19-s + 0.537·20-s + 1.78·22-s − 0.925·23-s − 0.0614·24-s − 0.565·25-s + 0.479·27-s + 0.0311·29-s + 0.219·30-s − 0.942·31-s + 1.30·32-s + 0.327·33-s − 0.393·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2941746319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2941746319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 3 | \( 1 + 0.428T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 17 | \( 1 - 1.20T + 17T^{2} \) |
| 19 | \( 1 + 3.24T + 19T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 - 0.167T + 29T^{2} \) |
| 31 | \( 1 + 5.24T + 31T^{2} \) |
| 37 | \( 1 - 7.05T + 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 - 0.0227T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 0.141T + 53T^{2} \) |
| 59 | \( 1 - 5.34T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 + 9.96T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 0.774T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 6.55T + 89T^{2} \) |
| 97 | \( 1 + 3.49T + 97T^{2} \) |
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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
−7.961571011622999994386850376324, −7.43556930329436879496381620037, −6.43527565538920545260867809302, −5.88063971259177853238558962960, −5.20386036664322956315803183187, −4.40280630804853183646582096675, −3.18742032222801843492213116359, −2.31295052864149071679363394440, −1.68645585870131622499901142520, −0.32055482015236223774701580294,
0.32055482015236223774701580294, 1.68645585870131622499901142520, 2.31295052864149071679363394440, 3.18742032222801843492213116359, 4.40280630804853183646582096675, 5.20386036664322956315803183187, 5.88063971259177853238558962960, 6.43527565538920545260867809302, 7.43556930329436879496381620037, 7.961571011622999994386850376324
