| L(s) = 1 | − 234·2-s + 2.18e3·3-s + 2.19e4·4-s − 2.80e5·5-s − 5.11e5·6-s + 2.52e6·8-s + 4.78e6·9-s + 6.56e7·10-s + 3.40e7·11-s + 4.80e7·12-s − 3.84e8·13-s − 6.13e8·15-s − 1.31e9·16-s − 1.25e9·17-s − 1.11e9·18-s + 2.49e9·19-s − 6.17e9·20-s − 7.96e9·22-s + 1.12e10·23-s + 5.51e9·24-s + 4.82e10·25-s + 8.98e10·26-s + 1.04e10·27-s − 4.84e10·29-s + 1.43e11·30-s − 1.30e11·31-s + 2.24e11·32-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.577·3-s + 0.671·4-s − 1.60·5-s − 0.746·6-s + 0.425·8-s + 1/3·9-s + 2.07·10-s + 0.526·11-s + 0.387·12-s − 1.69·13-s − 0.927·15-s − 1.22·16-s − 0.744·17-s − 0.430·18-s + 0.641·19-s − 1.07·20-s − 0.680·22-s + 0.691·23-s + 0.245·24-s + 1.58·25-s + 2.19·26-s + 0.192·27-s − 0.521·29-s + 1.19·30-s − 0.852·31-s + 1.15·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.2674269534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2674269534\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{7} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 117 p T + p^{15} T^{2} \) |
| 5 | \( 1 + 56142 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 3093732 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 29540174 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 1259207586 T + p^{15} T^{2} \) |
| 19 | \( 1 - 2499071020 T + p^{15} T^{2} \) |
| 23 | \( 1 - 11284833672 T + p^{15} T^{2} \) |
| 29 | \( 1 + 48413458530 T + p^{15} T^{2} \) |
| 31 | \( 1 + 130547265752 T + p^{15} T^{2} \) |
| 37 | \( 1 + 200223317554 T + p^{15} T^{2} \) |
| 41 | \( 1 + 679141724202 T + p^{15} T^{2} \) |
| 43 | \( 1 - 279482194892 T + p^{15} T^{2} \) |
| 47 | \( 1 + 1520672832576 T + p^{15} T^{2} \) |
| 53 | \( 1 - 2646053822502 T + p^{15} T^{2} \) |
| 59 | \( 1 + 7399371294540 T + p^{15} T^{2} \) |
| 61 | \( 1 - 42659617819498 T + p^{15} T^{2} \) |
| 67 | \( 1 + 56408026065964 T + p^{15} T^{2} \) |
| 71 | \( 1 + 133149677299848 T + p^{15} T^{2} \) |
| 73 | \( 1 + 105603350884922 T + p^{15} T^{2} \) |
| 79 | \( 1 + 55665674361880 T + p^{15} T^{2} \) |
| 83 | \( 1 + 378077412997332 T + p^{15} T^{2} \) |
| 89 | \( 1 + 219315065897610 T + p^{15} T^{2} \) |
| 97 | \( 1 + 703322682162626 T + p^{15} T^{2} \) |
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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
−10.06805948489692096722900159594, −9.133779067723181379509446220205, −8.431447309954482086709219327107, −7.31032722233198681149256064607, −7.20643575579388110872265503460, −4.86116441329120510470395337973, −3.96792275652468273637072536548, −2.75581997903107221087140825351, −1.48308513434377042023324580691, −0.26284475409824181448338049370,
0.26284475409824181448338049370, 1.48308513434377042023324580691, 2.75581997903107221087140825351, 3.96792275652468273637072536548, 4.86116441329120510470395337973, 7.20643575579388110872265503460, 7.31032722233198681149256064607, 8.431447309954482086709219327107, 9.133779067723181379509446220205, 10.06805948489692096722900159594
