| L(s) = 1 | + 6.40·2-s − 15.0·3-s + 9.01·4-s + 25·5-s − 96.4·6-s − 122.·7-s − 147.·8-s − 16.0·9-s + 160.·10-s + 121·11-s − 135.·12-s − 1.04e3·13-s − 783.·14-s − 376.·15-s − 1.23e3·16-s + 400.·17-s − 102.·18-s + 581.·19-s + 225.·20-s + 1.84e3·21-s + 774.·22-s + 66.9·23-s + 2.21e3·24-s + 625·25-s − 6.67e3·26-s + 3.90e3·27-s − 1.10e3·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 0.966·3-s + 0.281·4-s + 0.447·5-s − 1.09·6-s − 0.944·7-s − 0.813·8-s − 0.0660·9-s + 0.506·10-s + 0.301·11-s − 0.272·12-s − 1.71·13-s − 1.06·14-s − 0.432·15-s − 1.20·16-s + 0.336·17-s − 0.0747·18-s + 0.369·19-s + 0.126·20-s + 0.912·21-s + 0.341·22-s + 0.0263·23-s + 0.785·24-s + 0.200·25-s − 1.93·26-s + 1.03·27-s − 0.266·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 6.40T + 32T^{2} \) |
| 3 | \( 1 + 15.0T + 243T^{2} \) |
| 7 | \( 1 + 122.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 400.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 581.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 66.9T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.45e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.85e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.73e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.46e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.99e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−13.72617401191530719454330205047, −12.37735047622502092734216901767, −12.05981789016524671527788381631, −10.36603159186804327308352374331, −9.198515378405727399374327801566, −6.87501437414190963423096779922, −5.79233580028495839362506662210, −4.81385584202623914801612656302, −2.98723004237040680542163932172, 0,
2.98723004237040680542163932172, 4.81385584202623914801612656302, 5.79233580028495839362506662210, 6.87501437414190963423096779922, 9.198515378405727399374327801566, 10.36603159186804327308352374331, 12.05981789016524671527788381631, 12.37735047622502092734216901767, 13.72617401191530719454330205047
