| L(s) = 1 | + 23.3·2-s + 296.·3-s − 1.50e3·4-s − 1.33e4·5-s + 6.90e3·6-s + 3.29e4·7-s − 8.28e4·8-s − 8.93e4·9-s − 3.10e5·10-s + 1.61e5·11-s − 4.45e5·12-s − 1.25e6·13-s + 7.67e5·14-s − 3.94e6·15-s + 1.15e6·16-s + 4.21e6·17-s − 2.08e6·18-s − 1.48e7·19-s + 2.00e7·20-s + 9.76e6·21-s + 3.75e6·22-s + 3.52e7·23-s − 2.45e7·24-s + 1.28e8·25-s − 2.93e7·26-s − 7.89e7·27-s − 4.95e7·28-s + ⋯ |
| L(s) = 1 | + 0.515·2-s + 0.703·3-s − 0.734·4-s − 1.90·5-s + 0.362·6-s + 0.740·7-s − 0.893·8-s − 0.504·9-s − 0.982·10-s + 0.301·11-s − 0.517·12-s − 0.940·13-s + 0.381·14-s − 1.34·15-s + 0.274·16-s + 0.720·17-s − 0.259·18-s − 1.37·19-s + 1.40·20-s + 0.521·21-s + 0.155·22-s + 1.14·23-s − 0.628·24-s + 2.63·25-s − 0.484·26-s − 1.05·27-s − 0.544·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 1.61e5T \) |
| good | 2 | \( 1 - 23.3T + 2.04e3T^{2} \) |
| 3 | \( 1 - 296.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.33e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.29e4T + 1.97e9T^{2} \) |
| 13 | \( 1 + 1.25e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.21e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.48e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.52e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.15e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 5.28e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 1.71e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.25e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.41e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.89e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 8.18e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + 8.91e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 8.43e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 3.86e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.51e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.31e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.72e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.35e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.67e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 2.79e10T + 7.15e21T^{2} \) |
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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
−16.99948334666755274151022923148, −14.93257418061147758179368688434, −14.63051300288929024087883648916, −12.65337726312374367007575050239, −11.39452460469157226092504140927, −8.818359049979517687058909589608, −7.75624198653749582752056113611, −4.69980943478438159962829993669, −3.37928281104705399138313096130, 0,
3.37928281104705399138313096130, 4.69980943478438159962829993669, 7.75624198653749582752056113611, 8.818359049979517687058909589608, 11.39452460469157226092504140927, 12.65337726312374367007575050239, 14.63051300288929024087883648916, 14.93257418061147758179368688434, 16.99948334666755274151022923148
