| L(s) = 1 | − 4.85·2-s + 1.14·3-s + 15.5·4-s − 5.55·6-s − 36.6·8-s − 25.6·9-s − 39.7·11-s + 17.8·12-s − 61.1·13-s + 53.3·16-s + 79.0·17-s + 124.·18-s − 2.34·19-s + 193.·22-s + 186.·23-s − 41.9·24-s + 296.·26-s − 60.3·27-s + 62.5·29-s + 209.·31-s + 34.1·32-s − 45.5·33-s − 383.·34-s − 399.·36-s + 207.·37-s + 11.3·38-s − 69.9·39-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.220·3-s + 1.94·4-s − 0.378·6-s − 1.61·8-s − 0.951·9-s − 1.09·11-s + 0.428·12-s − 1.30·13-s + 0.833·16-s + 1.12·17-s + 1.63·18-s − 0.0283·19-s + 1.87·22-s + 1.69·23-s − 0.356·24-s + 2.23·26-s − 0.430·27-s + 0.400·29-s + 1.21·31-s + 0.188·32-s − 0.240·33-s − 1.93·34-s − 1.84·36-s + 0.921·37-s + 0.0485·38-s − 0.287·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.85T + 8T^{2} \) |
| 3 | \( 1 - 1.14T + 27T^{2} \) |
| 11 | \( 1 + 39.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 79.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.34T + 6.85e3T^{2} \) |
| 23 | \( 1 - 186.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 62.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 207.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 346.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 271.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 539.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 594.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 224.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 15.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 503.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 660.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 694.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 169.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 335.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 251.T + 9.12e5T^{2} \) |
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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
−8.949448301667780241006195106478, −8.157519068203922615184110365283, −7.64375146912999205480740284299, −6.89643185560589364388708662761, −5.73689829373809398823491858384, −4.83310597962734518841464030754, −2.93243600071866041523887407838, −2.52664614528137746450864436343, −1.03168606816760299413519921169, 0,
1.03168606816760299413519921169, 2.52664614528137746450864436343, 2.93243600071866041523887407838, 4.83310597962734518841464030754, 5.73689829373809398823491858384, 6.89643185560589364388708662761, 7.64375146912999205480740284299, 8.157519068203922615184110365283, 8.949448301667780241006195106478
