| L(s) = 1 | + 2.66·3-s + 2.69·5-s − 7·7-s − 19.9·9-s − 28.4·11-s + 38.0·13-s + 7.17·15-s + 97.3·17-s − 33.6·19-s − 18.6·21-s + 46.3·23-s − 117.·25-s − 124.·27-s + 18.5·29-s + 231.·31-s − 75.6·33-s − 18.8·35-s + 156.·37-s + 101.·39-s − 41·41-s − 379.·43-s − 53.6·45-s − 553.·47-s + 49·49-s + 259.·51-s + 208.·53-s − 76.5·55-s + ⋯ |
| L(s) = 1 | + 0.512·3-s + 0.240·5-s − 0.377·7-s − 0.737·9-s − 0.778·11-s + 0.811·13-s + 0.123·15-s + 1.38·17-s − 0.406·19-s − 0.193·21-s + 0.420·23-s − 0.941·25-s − 0.890·27-s + 0.118·29-s + 1.33·31-s − 0.399·33-s − 0.0910·35-s + 0.693·37-s + 0.416·39-s − 0.156·41-s − 1.34·43-s − 0.177·45-s − 1.71·47-s + 0.142·49-s + 0.711·51-s + 0.540·53-s − 0.187·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 - 2.66T + 27T^{2} \) |
| 5 | \( 1 - 2.69T + 125T^{2} \) |
| 11 | \( 1 + 28.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 97.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 46.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 18.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 231.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 156.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 379.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 553.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 208.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 851.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 839.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 77.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 818.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 479.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 974.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 339.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.40T + 9.12e5T^{2} \) |
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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
−8.951505958925877348812659860530, −8.181655292413973543370838411652, −7.62603954066570074753543161896, −6.30263287771581667467438290053, −5.77232787060598591089283156447, −4.68935606801305325388953865265, −3.37627184268896425224364212690, −2.83321219722653529705648634240, −1.49827186949034381682841050256, 0,
1.49827186949034381682841050256, 2.83321219722653529705648634240, 3.37627184268896425224364212690, 4.68935606801305325388953865265, 5.77232787060598591089283156447, 6.30263287771581667467438290053, 7.62603954066570074753543161896, 8.181655292413973543370838411652, 8.951505958925877348812659860530
