| L(s) = 1 | + 2-s − 3-s − 4-s + 2·5-s − 6-s − 2·7-s − 3·8-s − 2·9-s + 2·10-s + 12-s − 2·14-s − 2·15-s − 16-s − 7·17-s − 2·18-s − 6·19-s − 2·20-s + 2·21-s − 23-s + 3·24-s − 25-s + 5·27-s + 2·28-s + 3·29-s − 2·30-s − 10·31-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.755·7-s − 1.06·8-s − 2/3·9-s + 0.632·10-s + 0.288·12-s − 0.534·14-s − 0.516·15-s − 1/4·16-s − 1.69·17-s − 0.471·18-s − 1.37·19-s − 0.447·20-s + 0.436·21-s − 0.208·23-s + 0.612·24-s − 1/5·25-s + 0.962·27-s + 0.377·28-s + 0.557·29-s − 0.365·30-s − 1.79·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.04587291815915, −15.65480044318897, −14.91888606740595, −14.49117033386088, −13.78600110679779, −13.55428102177856, −12.86756850857429, −12.57557986674530, −12.00204224397909, −11.13537368480197, −10.81005050059432, −10.13472902106709, −9.409930585751689, −8.947761508402237, −8.631972462696028, −7.661631586075241, −6.521186155708332, −6.398410932791116, −5.917011817108814, −5.156814196276101, −4.694336545321085, −3.944471291947917, −3.219008277283329, −2.460738857049715, −1.723192366081595, 0, 0,
1.723192366081595, 2.460738857049715, 3.219008277283329, 3.944471291947917, 4.694336545321085, 5.156814196276101, 5.917011817108814, 6.398410932791116, 6.521186155708332, 7.661631586075241, 8.631972462696028, 8.947761508402237, 9.409930585751689, 10.13472902106709, 10.81005050059432, 11.13537368480197, 12.00204224397909, 12.57557986674530, 12.86756850857429, 13.55428102177856, 13.78600110679779, 14.49117033386088, 14.91888606740595, 15.65480044318897, 16.04587291815915
