| L(s) = 1 | − 2-s − 3.37·3-s + 4-s − 0.689·5-s + 3.37·6-s − 0.481·7-s − 8-s + 8.37·9-s + 0.689·10-s − 3.37·12-s + 3.54·13-s + 0.481·14-s + 2.32·15-s + 16-s − 5.33·17-s − 8.37·18-s + 19-s − 0.689·20-s + 1.62·21-s + 8.97·23-s + 3.37·24-s − 4.52·25-s − 3.54·26-s − 18.1·27-s − 0.481·28-s − 5.75·29-s − 2.32·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.94·3-s + 0.5·4-s − 0.308·5-s + 1.37·6-s − 0.181·7-s − 0.353·8-s + 2.79·9-s + 0.218·10-s − 0.973·12-s + 0.984·13-s + 0.128·14-s + 0.600·15-s + 0.250·16-s − 1.29·17-s − 1.97·18-s + 0.229·19-s − 0.154·20-s + 0.354·21-s + 1.87·23-s + 0.688·24-s − 0.904·25-s − 0.696·26-s − 3.48·27-s − 0.0909·28-s − 1.06·29-s − 0.424·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.37T + 3T^{2} \) |
| 5 | \( 1 + 0.689T + 5T^{2} \) |
| 7 | \( 1 + 0.481T + 7T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 + 5.33T + 17T^{2} \) |
| 23 | \( 1 - 8.97T + 23T^{2} \) |
| 29 | \( 1 + 5.75T + 29T^{2} \) |
| 31 | \( 1 + 3.56T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 41 | \( 1 - 2.97T + 41T^{2} \) |
| 43 | \( 1 + 2.54T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8.44T + 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 5.10T + 67T^{2} \) |
| 71 | \( 1 - 5.42T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 + 3.25T + 97T^{2} \) |
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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
−7.75171058010124720050697795559, −7.02147740674371052517356446588, −6.62057894991325335524737507260, −5.82402179586148627990232776181, −5.24123409052563820830538614905, −4.33272110649363469924689445590, −3.52716830869062250028397464713, −1.95054265283295669690845136375, −0.975594383811420601979321205447, 0,
0.975594383811420601979321205447, 1.95054265283295669690845136375, 3.52716830869062250028397464713, 4.33272110649363469924689445590, 5.24123409052563820830538614905, 5.82402179586148627990232776181, 6.62057894991325335524737507260, 7.02147740674371052517356446588, 7.75171058010124720050697795559
