L(s) = 1 | − 2-s + 3-s + 4-s − 2.87·5-s − 6-s − 0.347·7-s − 8-s + 9-s + 2.87·10-s + 11-s + 12-s − 2.53·13-s + 0.347·14-s − 2.87·15-s + 16-s + 4.75·17-s − 18-s − 2.83·19-s − 2.87·20-s − 0.347·21-s − 22-s − 3.53·23-s − 24-s + 3.29·25-s + 2.53·26-s + 27-s − 0.347·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.28·5-s − 0.408·6-s − 0.131·7-s − 0.353·8-s + 0.333·9-s + 0.910·10-s + 0.301·11-s + 0.288·12-s − 0.702·13-s + 0.0928·14-s − 0.743·15-s + 0.250·16-s + 1.15·17-s − 0.235·18-s − 0.650·19-s − 0.643·20-s − 0.0757·21-s − 0.213·22-s − 0.736·23-s − 0.204·24-s + 0.658·25-s + 0.496·26-s + 0.192·27-s − 0.0656·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 0.347T + 7T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 19 | \( 1 + 2.83T + 19T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 - 1.90T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 2.20T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 0.781T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 0.532T + 79T^{2} \) |
| 83 | \( 1 + 9.62T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 7.80T + 97T^{2} \) |
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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
−8.026021737622905351765329021845, −7.66263901585514643269519273924, −6.87163496228250944200134569512, −6.12388651727663782495362409260, −4.89648016076150829914454199365, −4.10608850285677161827857621597, −3.31746901204759266398855489828, −2.52881656452819256947787415540, −1.26428425200625308885363728382, 0,
1.26428425200625308885363728382, 2.52881656452819256947787415540, 3.31746901204759266398855489828, 4.10608850285677161827857621597, 4.89648016076150829914454199365, 6.12388651727663782495362409260, 6.87163496228250944200134569512, 7.66263901585514643269519273924, 8.026021737622905351765329021845