| L(s) = 1 | − 1.61·2-s − 2.61·3-s + 0.618·4-s + 2.23·5-s + 4.23·6-s + 0.618·7-s + 2.23·8-s + 3.85·9-s − 3.61·10-s − 4.85·11-s − 1.61·12-s − 13-s − 1.00·14-s − 5.85·15-s − 4.85·16-s − 17-s − 6.23·18-s − 6.85·19-s + 1.38·20-s − 1.61·21-s + 7.85·22-s + 5.23·23-s − 5.85·24-s + 1.61·26-s − 2.23·27-s + 0.381·28-s − 6.23·29-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 1.51·3-s + 0.309·4-s + 0.999·5-s + 1.72·6-s + 0.233·7-s + 0.790·8-s + 1.28·9-s − 1.14·10-s − 1.46·11-s − 0.467·12-s − 0.277·13-s − 0.267·14-s − 1.51·15-s − 1.21·16-s − 0.242·17-s − 1.46·18-s − 1.57·19-s + 0.309·20-s − 0.353·21-s + 1.67·22-s + 1.09·23-s − 1.19·24-s + 0.317·26-s − 0.430·27-s + 0.0721·28-s − 1.15·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 19 | \( 1 + 6.85T + 19T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 0.527T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 + 1.85T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 3.47T + 79T^{2} \) |
| 83 | \( 1 + 1.76T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−11.25458537692276774911581140601, −10.72408538021514946798233606800, −10.04615398343545307095761689964, −9.046254670461198105102031301030, −7.81929099927458841193439118383, −6.69881262089939985623785387713, −5.55682417528958702711193865772, −4.76726281932362069264088656976, −1.94369802528838451698226347566, 0,
1.94369802528838451698226347566, 4.76726281932362069264088656976, 5.55682417528958702711193865772, 6.69881262089939985623785387713, 7.81929099927458841193439118383, 9.046254670461198105102031301030, 10.04615398343545307095761689964, 10.72408538021514946798233606800, 11.25458537692276774911581140601
