| L(s) = 1 | − 8.43·3-s + 12.2·5-s − 1.63·7-s + 44.1·9-s + 25.7·11-s − 13.2·13-s − 103.·15-s − 53.6·17-s − 100.·19-s + 13.8·21-s − 25.1·23-s + 25.6·25-s − 144.·27-s + 256.·29-s − 132.·31-s − 217.·33-s − 20.1·35-s + 247.·37-s + 111.·39-s + 198.·41-s − 404.·43-s + 542.·45-s + 78.3·47-s − 340.·49-s + 452.·51-s + 743.·53-s + 315.·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 1.09·5-s − 0.0885·7-s + 1.63·9-s + 0.705·11-s − 0.282·13-s − 1.78·15-s − 0.764·17-s − 1.21·19-s + 0.143·21-s − 0.227·23-s + 0.205·25-s − 1.03·27-s + 1.63·29-s − 0.768·31-s − 1.14·33-s − 0.0971·35-s + 1.09·37-s + 0.457·39-s + 0.756·41-s − 1.43·43-s + 1.79·45-s + 0.243·47-s − 0.992·49-s + 1.24·51-s + 1.92·53-s + 0.774·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 8.43T + 27T^{2} \) |
| 5 | \( 1 - 12.2T + 125T^{2} \) |
| 7 | \( 1 + 1.63T + 343T^{2} \) |
| 11 | \( 1 - 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 100.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 256.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 247.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 404.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 78.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 743.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 65.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 273.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 399.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 727.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 106.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 58.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 580.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 768.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 809.T + 9.12e5T^{2} \) |
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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
−9.409821829532703112665407160773, −8.426112749048617706972910508705, −6.99394775701260236299647176170, −6.38407536534142023700988049736, −5.88653132612017082634067307679, −4.92530422849862585116143340037, −4.15168781600428229707176891025, −2.37554241051811622092092400429, −1.26587496214482953470862746498, 0,
1.26587496214482953470862746498, 2.37554241051811622092092400429, 4.15168781600428229707176891025, 4.92530422849862585116143340037, 5.88653132612017082634067307679, 6.38407536534142023700988049736, 6.99394775701260236299647176170, 8.426112749048617706972910508705, 9.409821829532703112665407160773
