| L(s) = 1 | + 2-s − 1.31·3-s + 4-s + 1.46·5-s − 1.31·6-s − 4.53·7-s + 8-s − 1.26·9-s + 1.46·10-s − 11-s − 1.31·12-s + 4.66·13-s − 4.53·14-s − 1.92·15-s + 16-s + 2.59·17-s − 1.26·18-s − 0.0248·19-s + 1.46·20-s + 5.97·21-s − 22-s + 3.39·23-s − 1.31·24-s − 2.86·25-s + 4.66·26-s + 5.61·27-s − 4.53·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.761·3-s + 0.5·4-s + 0.654·5-s − 0.538·6-s − 1.71·7-s + 0.353·8-s − 0.420·9-s + 0.462·10-s − 0.301·11-s − 0.380·12-s + 1.29·13-s − 1.21·14-s − 0.497·15-s + 0.250·16-s + 0.630·17-s − 0.297·18-s − 0.00569·19-s + 0.327·20-s + 1.30·21-s − 0.213·22-s + 0.708·23-s − 0.269·24-s − 0.572·25-s + 0.915·26-s + 1.08·27-s − 0.857·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 1.31T + 3T^{2} \) |
| 5 | \( 1 - 1.46T + 5T^{2} \) |
| 7 | \( 1 + 4.53T + 7T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 19 | \( 1 + 0.0248T + 19T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 29 | \( 1 + 3.84T + 29T^{2} \) |
| 31 | \( 1 + 3.79T + 31T^{2} \) |
| 37 | \( 1 + 2.61T + 37T^{2} \) |
| 41 | \( 1 - 5.88T + 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 + 1.04T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 7.61T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 + 6.22T + 73T^{2} \) |
| 79 | \( 1 + 3.75T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−7.81889810118770239413162492729, −6.92894747600202830095844871990, −6.22684701339927408221078345364, −5.84215347644855335797895139780, −5.39400366920773629377275326165, −4.18429628424725549609029937900, −3.31419769738336750398156458189, −2.78250422561938844357206905982, −1.40354077496965689986722631359, 0,
1.40354077496965689986722631359, 2.78250422561938844357206905982, 3.31419769738336750398156458189, 4.18429628424725549609029937900, 5.39400366920773629377275326165, 5.84215347644855335797895139780, 6.22684701339927408221078345364, 6.92894747600202830095844871990, 7.81889810118770239413162492729
