| L(s) = 1 | − 4.50·2-s − 15.5i·3-s − 43.6·4-s − 62.2i·5-s + 70.2i·6-s + 485.·8-s − 243·9-s + 280. i·10-s + 19.4·11-s + 681. i·12-s + 1.64e3i·13-s − 970.·15-s + 609.·16-s − 214. i·17-s + 1.09e3·18-s + 9.51e3i·19-s + ⋯ |
| L(s) = 1 | − 0.563·2-s − 0.577i·3-s − 0.682·4-s − 0.498i·5-s + 0.325i·6-s + 0.947·8-s − 0.333·9-s + 0.280i·10-s + 0.0146·11-s + 0.394i·12-s + 0.747i·13-s − 0.287·15-s + 0.148·16-s − 0.0437i·17-s + 0.187·18-s + 1.38i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.052878717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.052878717\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 15.5iT \) |
| 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
−12.02320743990018594268804686409, −10.56956285973985047679439346320, −9.596831189372235811828429248328, −8.567640497051790467033047879218, −7.891735208614047594336063063757, −6.52626166370149133704223524482, −5.13894737643161545242663770052, −3.89917273022330177069714108025, −1.85337082084089198580177492194, −0.64583916330615983736624361919,
0.75192436950476899714843598649, 2.80620835395418508970900204930, 4.23022225049939387197752930316, 5.31981613514309168110076982702, 6.85654957211853108794842764238, 8.171889193573792479360775647095, 8.975109539064079814812588286604, 10.13924869740856199126069064542, 10.60193542456632191431121423108, 11.91389004259280154897043622947
