| L(s) = 1 | + 0.567·3-s − 5-s − 2.93·7-s − 2.67·9-s + 4.64·11-s − 0.911·13-s − 0.567·15-s + 3.78·19-s − 1.66·21-s − 2.25·23-s + 25-s − 3.22·27-s + 1.11·29-s + 0.114·31-s + 2.63·33-s + 2.93·35-s + 10.1·37-s − 0.516·39-s − 0.709·41-s − 1.49·43-s + 2.67·45-s + 4.65·47-s + 1.63·49-s + 4.81·53-s − 4.64·55-s + 2.14·57-s − 2.55·59-s + ⋯ |
| L(s) = 1 | + 0.327·3-s − 0.447·5-s − 1.11·7-s − 0.892·9-s + 1.40·11-s − 0.252·13-s − 0.146·15-s + 0.868·19-s − 0.363·21-s − 0.469·23-s + 0.200·25-s − 0.619·27-s + 0.206·29-s + 0.0205·31-s + 0.458·33-s + 0.496·35-s + 1.66·37-s − 0.0827·39-s − 0.110·41-s − 0.227·43-s + 0.399·45-s + 0.678·47-s + 0.233·49-s + 0.661·53-s − 0.626·55-s + 0.284·57-s − 0.332·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 0.567T + 3T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 + 0.911T + 13T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 - 1.11T + 29T^{2} \) |
| 31 | \( 1 - 0.114T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 0.709T + 41T^{2} \) |
| 43 | \( 1 + 1.49T + 43T^{2} \) |
| 47 | \( 1 - 4.65T + 47T^{2} \) |
| 53 | \( 1 - 4.81T + 53T^{2} \) |
| 59 | \( 1 + 2.55T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 - 2.81T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.66335331677366442299064837299, −7.15581677129595871393761553174, −6.15070533513134119673125360565, −5.95724054815804761584479051461, −4.70515671961406928113943369923, −3.91220893981743279903363224156, −3.23447574850783657040235973113, −2.60464005131366929657748599026, −1.24535511833096681409446023975, 0,
1.24535511833096681409446023975, 2.60464005131366929657748599026, 3.23447574850783657040235973113, 3.91220893981743279903363224156, 4.70515671961406928113943369923, 5.95724054815804761584479051461, 6.15070533513134119673125360565, 7.15581677129595871393761553174, 7.66335331677366442299064837299
