| L(s) = 1 | − 3-s − 0.801·5-s − 1.69·7-s + 9-s + 1.24·11-s + 0.801·15-s − 0.445·17-s + 2.04·19-s + 1.69·21-s + 0.801·23-s − 4.35·25-s − 27-s − 2.46·29-s − 1.57·31-s − 1.24·33-s + 1.35·35-s + 9.54·37-s + 7.56·41-s − 0.286·43-s − 0.801·45-s + 0.542·47-s − 4.13·49-s + 0.445·51-s − 2.45·53-s − 55-s − 2.04·57-s − 14.3·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.358·5-s − 0.639·7-s + 0.333·9-s + 0.375·11-s + 0.207·15-s − 0.107·17-s + 0.470·19-s + 0.369·21-s + 0.167·23-s − 0.871·25-s − 0.192·27-s − 0.458·29-s − 0.283·31-s − 0.217·33-s + 0.229·35-s + 1.56·37-s + 1.18·41-s − 0.0436·43-s − 0.119·45-s + 0.0791·47-s − 0.591·49-s + 0.0623·51-s − 0.337·53-s − 0.134·55-s − 0.271·57-s − 1.87·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.801T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 17 | \( 1 + 0.445T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 - 0.801T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 - 9.54T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 + 0.286T + 43T^{2} \) |
| 47 | \( 1 - 0.542T + 47T^{2} \) |
| 53 | \( 1 + 2.45T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 2.81T + 67T^{2} \) |
| 71 | \( 1 - 6.96T + 71T^{2} \) |
| 73 | \( 1 - 3.69T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 4.55T + 83T^{2} \) |
| 89 | \( 1 - 8.02T + 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−7.83609608467465353143969559690, −7.42370818098007421402746434076, −6.43205342280761683144083580236, −6.01253231501341882870053191767, −5.10105905588717324366698752815, −4.22106337310495962484251512269, −3.54268439894040048230514802429, −2.51142751478136838650700321647, −1.22981524761046280064540520814, 0,
1.22981524761046280064540520814, 2.51142751478136838650700321647, 3.54268439894040048230514802429, 4.22106337310495962484251512269, 5.10105905588717324366698752815, 6.01253231501341882870053191767, 6.43205342280761683144083580236, 7.42370818098007421402746434076, 7.83609608467465353143969559690
