| L(s) = 1 | − 2·4-s + 3.46·5-s − 1.73·7-s − 3.46·11-s + 4·16-s − 3.46·19-s − 6.92·20-s − 6·23-s + 6.99·25-s + 3.46·28-s − 6·29-s − 1.73·31-s − 5.99·35-s + 6.92·41-s + 43-s + 6.92·44-s − 3.46·47-s − 4·49-s − 12·53-s − 11.9·55-s + 3.46·59-s + 61-s − 8·64-s + 8.66·67-s − 10.3·71-s + 1.73·73-s + 6.92·76-s + ⋯ |
| L(s) = 1 | − 4-s + 1.54·5-s − 0.654·7-s − 1.04·11-s + 16-s − 0.794·19-s − 1.54·20-s − 1.25·23-s + 1.39·25-s + 0.654·28-s − 1.11·29-s − 0.311·31-s − 1.01·35-s + 1.08·41-s + 0.152·43-s + 1.04·44-s − 0.505·47-s − 0.571·49-s − 1.64·53-s − 1.61·55-s + 0.450·59-s + 0.128·61-s − 64-s + 1.05·67-s − 1.23·71-s + 0.202·73-s + 0.794·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 8.66T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−9.263350720538986503949105687311, −8.433072077157405642270976195742, −7.56773026814495758560850966451, −6.31271200054070109359894624843, −5.78042354082766657034611225905, −5.04981090419579758935387031604, −4.00306439752385457084057769673, −2.82072525284652420726382253881, −1.77572593776434704193686502476, 0,
1.77572593776434704193686502476, 2.82072525284652420726382253881, 4.00306439752385457084057769673, 5.04981090419579758935387031604, 5.78042354082766657034611225905, 6.31271200054070109359894624843, 7.56773026814495758560850966451, 8.433072077157405642270976195742, 9.263350720538986503949105687311
