| L(s) = 1 | − 2.21·3-s + 0.903·7-s + 1.90·9-s − 0.0666·11-s − 13-s − 3.37·17-s + 5.11·19-s − 2·21-s − 4.21·23-s + 2.42·27-s + 1.52·29-s + 4.49·31-s + 0.147·33-s − 11.9·37-s + 2.21·39-s + 2.75·41-s + 8.77·43-s − 8.90·47-s − 6.18·49-s + 7.47·51-s − 3.57·53-s − 11.3·57-s − 8.16·59-s − 11.1·61-s + 1.71·63-s − 2.14·67-s + 9.33·69-s + ⋯ |
| L(s) = 1 | − 1.27·3-s + 0.341·7-s + 0.634·9-s − 0.0201·11-s − 0.277·13-s − 0.819·17-s + 1.17·19-s − 0.436·21-s − 0.878·23-s + 0.467·27-s + 0.283·29-s + 0.807·31-s + 0.0257·33-s − 1.96·37-s + 0.354·39-s + 0.430·41-s + 1.33·43-s − 1.29·47-s − 0.883·49-s + 1.04·51-s − 0.490·53-s − 1.50·57-s − 1.06·59-s − 1.43·61-s + 0.216·63-s − 0.262·67-s + 1.12·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 7 | \( 1 - 0.903T + 7T^{2} \) |
| 11 | \( 1 + 0.0666T + 11T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 - 5.11T + 19T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 - 4.49T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 2.75T + 41T^{2} \) |
| 43 | \( 1 - 8.77T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 + 3.57T + 53T^{2} \) |
| 59 | \( 1 + 8.16T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 2.14T + 67T^{2} \) |
| 71 | \( 1 + 5.54T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 + 3.18T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 7.93T + 97T^{2} \) |
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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
−9.380893441586795737648248475385, −8.393830419625992227921342981403, −7.48637646375233336387122158236, −6.61819878228298905940589960575, −5.88004864630959631449936914993, −5.06084906891072844753638401045, −4.37342385627173188163923632525, −2.98481710563334863009165194968, −1.50843715512195702612944265528, 0,
1.50843715512195702612944265528, 2.98481710563334863009165194968, 4.37342385627173188163923632525, 5.06084906891072844753638401045, 5.88004864630959631449936914993, 6.61819878228298905940589960575, 7.48637646375233336387122158236, 8.393830419625992227921342981403, 9.380893441586795737648248475385
