| L(s) = 1 | − 2.56·3-s − 4.56·7-s + 3.56·9-s − 11-s + 1.12·13-s + 7.68·17-s + 1.43·19-s + 11.6·21-s − 1.12·23-s − 1.43·27-s + 8.56·29-s − 1.43·31-s + 2.56·33-s − 7.43·37-s − 2.87·39-s − 12.2·41-s + 3.12·43-s + 11.3·47-s + 13.8·49-s − 19.6·51-s − 9.68·53-s − 3.68·57-s + 1.12·59-s − 12.5·61-s − 16.2·63-s + 2.87·69-s + 3.68·71-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 1.72·7-s + 1.18·9-s − 0.301·11-s + 0.311·13-s + 1.86·17-s + 0.330·19-s + 2.54·21-s − 0.234·23-s − 0.276·27-s + 1.58·29-s − 0.258·31-s + 0.445·33-s − 1.22·37-s − 0.460·39-s − 1.91·41-s + 0.476·43-s + 1.65·47-s + 1.97·49-s − 2.75·51-s − 1.33·53-s − 0.488·57-s + 0.146·59-s − 1.60·61-s − 2.04·63-s + 0.346·69-s + 0.437·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 13 | \( 1 - 1.12T + 13T^{2} \) |
| 17 | \( 1 - 7.68T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 + 1.43T + 31T^{2} \) |
| 37 | \( 1 + 7.43T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 1.12T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 9.68T + 89T^{2} \) |
| 97 | \( 1 - 4.87T + 97T^{2} \) |
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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
−8.785402291776056964809231708808, −7.67172082986766673362073512417, −6.84440514355326304137414267431, −6.22890009525466020544546426678, −5.64324103213377772433090446760, −4.91771582944466859018289888002, −3.65716234947209555278738095568, −2.95911103601770103329387322429, −1.14086041607664668277828186316, 0,
1.14086041607664668277828186316, 2.95911103601770103329387322429, 3.65716234947209555278738095568, 4.91771582944466859018289888002, 5.64324103213377772433090446760, 6.22890009525466020544546426678, 6.84440514355326304137414267431, 7.67172082986766673362073512417, 8.785402291776056964809231708808
