| L(s) = 1 | − 3·3-s + 2·5-s + 6·9-s − 2·11-s + 5·13-s − 6·15-s − 4·17-s − 2·19-s − 23-s − 25-s − 9·27-s − 7·29-s − 3·31-s + 6·33-s + 2·37-s − 15·39-s + 9·41-s + 8·43-s + 12·45-s + 9·47-s + 12·51-s + 2·53-s − 4·55-s + 6·57-s + 2·61-s + 10·65-s − 14·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s + 2·9-s − 0.603·11-s + 1.38·13-s − 1.54·15-s − 0.970·17-s − 0.458·19-s − 0.208·23-s − 1/5·25-s − 1.73·27-s − 1.29·29-s − 0.538·31-s + 1.04·33-s + 0.328·37-s − 2.40·39-s + 1.40·41-s + 1.21·43-s + 1.78·45-s + 1.31·47-s + 1.68·51-s + 0.274·53-s − 0.539·55-s + 0.794·57-s + 0.256·61-s + 1.24·65-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−16.27909887150650, −15.63652052069520, −15.26564541484603, −14.37818901946680, −13.59033445393998, −13.26892060690725, −12.77039123897024, −12.23861847984697, −11.45893656370887, −10.89368473925218, −10.80310848050200, −10.16180547422278, −9.293856887965054, −9.026254450057177, −7.971553012205529, −7.342931016587648, −6.586449427051673, −6.078484785841092, −5.697368306761710, −5.261931075944583, −4.252820034584717, −3.946275098151477, −2.564520117444572, −1.819395854862916, −0.9830644877250290, 0,
0.9830644877250290, 1.819395854862916, 2.564520117444572, 3.946275098151477, 4.252820034584717, 5.261931075944583, 5.697368306761710, 6.078484785841092, 6.586449427051673, 7.342931016587648, 7.971553012205529, 9.026254450057177, 9.293856887965054, 10.16180547422278, 10.80310848050200, 10.89368473925218, 11.45893656370887, 12.23861847984697, 12.77039123897024, 13.26892060690725, 13.59033445393998, 14.37818901946680, 15.26564541484603, 15.63652052069520, 16.27909887150650
