| L(s) = 1 | − 2-s − 3·3-s − 4-s + 3·6-s − 4·7-s + 3·8-s + 6·9-s + 3·12-s − 3·13-s + 4·14-s − 16-s − 3·17-s − 6·18-s − 5·19-s + 12·21-s + 7·23-s − 9·24-s − 5·25-s + 3·26-s − 9·27-s + 4·28-s − 7·29-s + 4·31-s − 5·32-s + 3·34-s − 6·36-s + 5·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.22·6-s − 1.51·7-s + 1.06·8-s + 2·9-s + 0.866·12-s − 0.832·13-s + 1.06·14-s − 1/4·16-s − 0.727·17-s − 1.41·18-s − 1.14·19-s + 2.61·21-s + 1.45·23-s − 1.83·24-s − 25-s + 0.588·26-s − 1.73·27-s + 0.755·28-s − 1.29·29-s + 0.718·31-s − 0.883·32-s + 0.514·34-s − 36-s + 0.821·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.32142426611890570761423436295, −13.21929603222721598677265581144, −12.62142615234456554387532222483, −11.17082899287976457738206329859, −10.14893340042306283098541226584, −9.264238826193943461669666199356, −7.20736947999270954805514319970, −6.04347889941024738587947019148, −4.50862835068296220400083952922, 0,
4.50862835068296220400083952922, 6.04347889941024738587947019148, 7.20736947999270954805514319970, 9.264238826193943461669666199356, 10.14893340042306283098541226584, 11.17082899287976457738206329859, 12.62142615234456554387532222483, 13.21929603222721598677265581144, 15.32142426611890570761423436295
