| L(s) = 1 | + 2-s + 4-s − 2.11·5-s − 4.03·7-s + 8-s − 2.11·10-s + 3.15·13-s − 4.03·14-s + 16-s + 4.88·19-s − 2.11·20-s − 0.887·23-s − 0.537·25-s + 3.15·26-s − 4.03·28-s + 4.80·29-s − 2.65·31-s + 32-s + 8.53·35-s + 3.19·37-s + 4.88·38-s − 2.11·40-s + 6.99·41-s − 12.9·43-s − 0.887·46-s − 1.85·47-s + 9.31·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.944·5-s − 1.52·7-s + 0.353·8-s − 0.668·10-s + 0.874·13-s − 1.07·14-s + 0.250·16-s + 1.12·19-s − 0.472·20-s − 0.185·23-s − 0.107·25-s + 0.618·26-s − 0.763·28-s + 0.891·29-s − 0.477·31-s + 0.176·32-s + 1.44·35-s + 0.525·37-s + 0.792·38-s − 0.334·40-s + 1.09·41-s − 1.98·43-s − 0.130·46-s − 0.271·47-s + 1.33·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 + 4.03T + 7T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 4.88T + 19T^{2} \) |
| 23 | \( 1 + 0.887T + 23T^{2} \) |
| 29 | \( 1 - 4.80T + 29T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 - 6.99T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + 1.85T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 + 8.23T + 59T^{2} \) |
| 61 | \( 1 + 7.22T + 61T^{2} \) |
| 67 | \( 1 - 2.80T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 7.81T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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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
−7.41578280056964263172716267996, −6.96698188475035982602155675785, −6.11982546446734611140037781336, −5.69819165905742601781671425694, −4.60219947740652771973792664545, −3.88893317059303777195448495558, −3.30848790371782520459178081967, −2.74737784529083684110692806176, −1.27174829045552710909662417073, 0,
1.27174829045552710909662417073, 2.74737784529083684110692806176, 3.30848790371782520459178081967, 3.88893317059303777195448495558, 4.60219947740652771973792664545, 5.69819165905742601781671425694, 6.11982546446734611140037781336, 6.96698188475035982602155675785, 7.41578280056964263172716267996
