| L(s) = 1 | − 1.78·2-s − 2.32·3-s + 1.20·4-s + 2.32·5-s + 4.15·6-s + 1.42·8-s + 2.38·9-s − 4.15·10-s + 2.11·11-s − 2.78·12-s + 2.32·13-s − 5.38·15-s − 4.95·16-s − 1.24·17-s − 4.26·18-s − 2.43·19-s + 2.78·20-s − 3.77·22-s − 5.86·23-s − 3.31·24-s + 0.385·25-s − 4.15·26-s + 1.42·27-s − 6.48·29-s + 9.63·30-s + 2.53·31-s + 6.01·32-s + ⋯ |
| L(s) = 1 | − 1.26·2-s − 1.33·3-s + 0.600·4-s + 1.03·5-s + 1.69·6-s + 0.504·8-s + 0.795·9-s − 1.31·10-s + 0.636·11-s − 0.805·12-s + 0.643·13-s − 1.39·15-s − 1.23·16-s − 0.301·17-s − 1.00·18-s − 0.559·19-s + 0.623·20-s − 0.804·22-s − 1.22·23-s − 0.676·24-s + 0.0770·25-s − 0.814·26-s + 0.274·27-s − 1.20·29-s + 1.75·30-s + 0.454·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.78T + 2T^{2} \) |
| 3 | \( 1 + 2.32T + 3T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 11 | \( 1 - 2.11T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 + 0.401T + 37T^{2} \) |
| 43 | \( 1 + 0.872T + 43T^{2} \) |
| 47 | \( 1 + 3.98T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 - 6.44T + 73T^{2} \) |
| 79 | \( 1 - 4.18T + 79T^{2} \) |
| 83 | \( 1 - 4.83T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.962145962934026292815652959937, −8.144324152027442383842587531273, −7.14805481575746423242160159419, −6.27920687145495336603577569425, −5.93239748495153862544582967051, −4.90589329441035949389123871686, −3.94280425499484227491364764445, −2.12400989936957542581865185596, −1.28590081289234090526552364340, 0,
1.28590081289234090526552364340, 2.12400989936957542581865185596, 3.94280425499484227491364764445, 4.90589329441035949389123871686, 5.93239748495153862544582967051, 6.27920687145495336603577569425, 7.14805481575746423242160159419, 8.144324152027442383842587531273, 8.962145962934026292815652959937
