| L(s) = 1 | + 1.89·2-s + 1.57·3-s + 1.59·4-s − 3.12·5-s + 2.99·6-s − 0.774·8-s − 0.510·9-s − 5.91·10-s − 2.47·11-s + 2.51·12-s + 3.58·13-s − 4.92·15-s − 4.65·16-s − 7.11·17-s − 0.966·18-s − 5.99·19-s − 4.96·20-s − 4.69·22-s + 9.03·23-s − 1.22·24-s + 4.74·25-s + 6.79·26-s − 5.53·27-s + 9.03·29-s − 9.33·30-s − 8.92·31-s − 7.26·32-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.910·3-s + 0.795·4-s − 1.39·5-s + 1.22·6-s − 0.273·8-s − 0.170·9-s − 1.87·10-s − 0.746·11-s + 0.724·12-s + 0.995·13-s − 1.27·15-s − 1.16·16-s − 1.72·17-s − 0.227·18-s − 1.37·19-s − 1.11·20-s − 1.00·22-s + 1.88·23-s − 0.249·24-s + 0.949·25-s + 1.33·26-s − 1.06·27-s + 1.67·29-s − 1.70·30-s − 1.60·31-s − 1.28·32-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 | \( 1 - 1.89T + 2T^{2} \) |
| 3 | \( 1 - 1.57T + 3T^{2} \) |
| 5 | \( 1 + 3.12T + 5T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 + 7.11T + 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 23 | \( 1 - 9.03T + 23T^{2} \) |
| 29 | \( 1 - 9.03T + 29T^{2} \) |
| 31 | \( 1 + 8.92T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 + 1.53T + 47T^{2} \) |
| 53 | \( 1 + 0.436T + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 2.71T + 67T^{2} \) |
| 71 | \( 1 - 4.30T + 71T^{2} \) |
| 73 | \( 1 - 6.16T + 73T^{2} \) |
| 79 | \( 1 - 0.388T + 79T^{2} \) |
| 83 | \( 1 - 7.05T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.610365875638209235459166007859, −8.168517392870457690623732094936, −6.99078008277397669403337439553, −6.46497255142083860020272690349, −5.20960397171379545227981683485, −4.50973165596107417158162150554, −3.74089273290555850415445642598, −3.14782045749312083255213448492, −2.26180862057172186239781314081, 0,
2.26180862057172186239781314081, 3.14782045749312083255213448492, 3.74089273290555850415445642598, 4.50973165596107417158162150554, 5.20960397171379545227981683485, 6.46497255142083860020272690349, 6.99078008277397669403337439553, 8.168517392870457690623732094936, 8.610365875638209235459166007859
