| L(s) = 1 | − 1.48·2-s + 3-s + 0.201·4-s − 5-s − 1.48·6-s − 7-s + 2.66·8-s + 9-s + 1.48·10-s + 4.66·11-s + 0.201·12-s + 5.18·13-s + 1.48·14-s − 15-s − 4.36·16-s + 2.74·17-s − 1.48·18-s − 1.91·19-s − 0.201·20-s − 21-s − 6.92·22-s − 23-s + 2.66·24-s + 25-s − 7.68·26-s + 27-s − 0.201·28-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.577·3-s + 0.100·4-s − 0.447·5-s − 0.605·6-s − 0.377·7-s + 0.943·8-s + 0.333·9-s + 0.469·10-s + 1.40·11-s + 0.0582·12-s + 1.43·13-s + 0.396·14-s − 0.258·15-s − 1.09·16-s + 0.664·17-s − 0.349·18-s − 0.440·19-s − 0.0451·20-s − 0.218·21-s − 1.47·22-s − 0.208·23-s + 0.544·24-s + 0.200·25-s − 1.50·26-s + 0.192·27-s − 0.0381·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.233626759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233626759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 11 | \( 1 - 4.66T + 11T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 - 2.74T + 17T^{2} \) |
| 19 | \( 1 + 1.91T + 19T^{2} \) |
| 29 | \( 1 - 1.79T + 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 - 2.84T + 43T^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 + 1.91T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 - 1.05T + 61T^{2} \) |
| 67 | \( 1 - 5.26T + 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 6.27T + 79T^{2} \) |
| 83 | \( 1 + 0.648T + 83T^{2} \) |
| 89 | \( 1 + 0.0377T + 89T^{2} \) |
| 97 | \( 1 - 9.17T + 97T^{2} \) |
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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
−8.891019624728933838771196694173, −8.356251837099724290639072273898, −7.77508822384952234483393038216, −6.74595533719075328615157931384, −6.24758366190272624908135788378, −4.79651481435800979412639130864, −3.90866259226100913666587586284, −3.31748565780195356526496007443, −1.75190025011924329747734122059, −0.877411588179431494822045308638,
0.877411588179431494822045308638, 1.75190025011924329747734122059, 3.31748565780195356526496007443, 3.90866259226100913666587586284, 4.79651481435800979412639130864, 6.24758366190272624908135788378, 6.74595533719075328615157931384, 7.77508822384952234483393038216, 8.356251837099724290639072273898, 8.891019624728933838771196694173
