| L(s) = 1 | − 19.0·3-s − 210.·7-s + 121.·9-s + 18.2·11-s + 834.·13-s − 292.·17-s + 2.28e3·19-s + 4.01e3·21-s − 3.78e3·23-s + 2.31e3·27-s + 4.55e3·29-s + 1.45e3·31-s − 348.·33-s + 5.50e3·37-s − 1.59e4·39-s − 616.·41-s − 1.58e4·43-s + 2.91e4·47-s + 2.73e4·49-s + 5.58e3·51-s − 3.69e4·53-s − 4.36e4·57-s − 1.87e4·59-s + 3.30e4·61-s − 2.55e4·63-s − 5.59e3·67-s + 7.23e4·69-s + ⋯ |
| L(s) = 1 | − 1.22·3-s − 1.62·7-s + 0.500·9-s + 0.0455·11-s + 1.37·13-s − 0.245·17-s + 1.45·19-s + 1.98·21-s − 1.49·23-s + 0.612·27-s + 1.00·29-s + 0.272·31-s − 0.0557·33-s + 0.661·37-s − 1.67·39-s − 0.0573·41-s − 1.30·43-s + 1.92·47-s + 1.62·49-s + 0.300·51-s − 1.80·53-s − 1.78·57-s − 0.699·59-s + 1.13·61-s − 0.810·63-s − 0.152·67-s + 1.82·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 19.0T + 243T^{2} \) |
| 7 | \( 1 + 210.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 18.2T + 1.61e5T^{2} \) |
| 13 | \( 1 - 834.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 292.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 616.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.58e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.87e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.59e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.45e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.93e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.33e5T + 8.58e9T^{2} \) |
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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
−10.09710951014079753754519845092, −9.339820435153326561803156908369, −8.127188158768506057336392140763, −6.72371414635739374343850780064, −6.21717541522253512518479959779, −5.44420586462550845768383812233, −4.01881268875298274650527961123, −2.97204456456654716333945808421, −1.05519298212469748941221613777, 0,
1.05519298212469748941221613777, 2.97204456456654716333945808421, 4.01881268875298274650527961123, 5.44420586462550845768383812233, 6.21717541522253512518479959779, 6.72371414635739374343850780064, 8.127188158768506057336392140763, 9.339820435153326561803156908369, 10.09710951014079753754519845092
