| L(s) = 1 | + 4.50·2-s − 43.6·4-s − 62.2i·5-s − 485.·8-s − 280. i·10-s − 19.4·11-s − 1.64e3i·13-s + 609.·16-s − 214. i·17-s − 9.51e3i·19-s + 2.72e3i·20-s − 87.5·22-s + 1.44e4·23-s + 1.17e4·25-s − 7.40e3i·26-s + ⋯ |
| L(s) = 1 | + 0.563·2-s − 0.682·4-s − 0.498i·5-s − 0.947·8-s − 0.280i·10-s − 0.0146·11-s − 0.747i·13-s + 0.148·16-s − 0.0437i·17-s − 1.38i·19-s + 0.340i·20-s − 0.00822·22-s + 1.18·23-s + 0.751·25-s − 0.421i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.05852973880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05852973880\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.50T + 64T^{2} \) |
| 5 | \( 1 + 62.2iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 19.4T + 1.77e6T^{2} \) |
| 13 | \( 1 + 1.64e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 214. iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 9.51e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.44e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 4.30e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 9.00e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.31e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 7.37e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 4.76e3T + 6.32e9T^{2} \) |
| 47 | \( 1 - 7.36e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.38e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.26e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 4.04e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.19e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 3.50e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 2.01e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 3.95e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.32e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.30e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 6.62e5iT - 8.32e11T^{2} \) |
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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
−9.250977551967751506613234074624, −8.947370761097983439553192779512, −7.78566500528877778014440989413, −6.66268980781760345738574000163, −5.35859032530045299835685817671, −4.93979413796741697380222028477, −3.75219666873155465383453050553, −2.74610556395146957189742282025, −1.02998649439112328476287669546, −0.01210520175746826489713789535,
1.56940274990375224583655662522, 3.05810025211933542462026122091, 3.90049673412251623531839939293, 4.93428035457508985546716471609, 5.88618044852628893449970092842, 6.85197135676441358459819165642, 7.957515427909265662762455866867, 8.997174113941687111274528448974, 9.689674764660135335966627628498, 10.75066579851535655798916520078
