L(s) = 1 | − 2-s + 2.15·3-s + 4-s − 1.75·5-s − 2.15·6-s + 1.64·7-s − 8-s + 1.65·9-s + 1.75·10-s − 2.44·11-s + 2.15·12-s + 0.440·13-s − 1.64·14-s − 3.79·15-s + 16-s + 0.339·17-s − 1.65·18-s − 5.30·19-s − 1.75·20-s + 3.55·21-s + 2.44·22-s + 6.94·23-s − 2.15·24-s − 1.91·25-s − 0.440·26-s − 2.89·27-s + 1.64·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.24·3-s + 0.5·4-s − 0.785·5-s − 0.881·6-s + 0.622·7-s − 0.353·8-s + 0.552·9-s + 0.555·10-s − 0.736·11-s + 0.622·12-s + 0.122·13-s − 0.440·14-s − 0.978·15-s + 0.250·16-s + 0.0823·17-s − 0.390·18-s − 1.21·19-s − 0.392·20-s + 0.775·21-s + 0.520·22-s + 1.44·23-s − 0.440·24-s − 0.383·25-s − 0.0864·26-s − 0.557·27-s + 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 + 1.75T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 0.440T + 13T^{2} \) |
| 17 | \( 1 - 0.339T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 29 | \( 1 + 0.0979T + 29T^{2} \) |
| 31 | \( 1 + 6.51T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 + 8.14T + 43T^{2} \) |
| 47 | \( 1 - 2.17T + 47T^{2} \) |
| 53 | \( 1 + 0.899T + 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 + 4.27T + 67T^{2} \) |
| 71 | \( 1 + 2.46T + 71T^{2} \) |
| 73 | \( 1 - 2.24T + 73T^{2} \) |
| 79 | \( 1 + 5.31T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 8.42T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.216019730565020734442030000496, −7.60121462636157131939207992168, −7.07928000887879636052811697291, −5.96999917718999756504206049910, −4.94604392696976516074533769521, −4.05365806355000474514301799090, −3.21491962375518503131694246399, −2.45996921266189506225256844728, −1.55032385706797600952388146493, 0,
1.55032385706797600952388146493, 2.45996921266189506225256844728, 3.21491962375518503131694246399, 4.05365806355000474514301799090, 4.94604392696976516074533769521, 5.96999917718999756504206049910, 7.07928000887879636052811697291, 7.60121462636157131939207992168, 8.216019730565020734442030000496