| L(s) = 1 | − 27·3-s − 125·5-s − 1.60e3·7-s + 729·9-s + 2.20e3·11-s + 5.73e3·13-s + 3.37e3·15-s + 1.56e4·17-s + 1.96e4·19-s + 4.33e4·21-s + 2.85e4·23-s + 1.56e4·25-s − 1.96e4·27-s − 1.40e5·29-s + 2.91e5·31-s − 5.96e4·33-s + 2.00e5·35-s − 1.35e5·37-s − 1.54e5·39-s − 8.04e5·41-s − 7.21e5·43-s − 9.11e4·45-s + 8.02e5·47-s + 1.74e6·49-s − 4.22e5·51-s + 2.74e5·53-s − 2.76e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.76·7-s + 1/3·9-s + 0.500·11-s + 0.724·13-s + 0.258·15-s + 0.772·17-s + 0.657·19-s + 1.02·21-s + 0.488·23-s + 1/5·25-s − 0.192·27-s − 1.06·29-s + 1.75·31-s − 0.288·33-s + 0.790·35-s − 0.438·37-s − 0.418·39-s − 1.82·41-s − 1.38·43-s − 0.149·45-s + 1.12·47-s + 2.12·49-s − 0.446·51-s + 0.252·53-s − 0.223·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 + p^{3} T \) |
| good | 7 | \( 1 + 1604 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2208 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5738 T + p^{7} T^{2} \) |
| 17 | \( 1 - 15654 T + p^{7} T^{2} \) |
| 19 | \( 1 - 19660 T + p^{7} T^{2} \) |
| 23 | \( 1 - 28512 T + p^{7} T^{2} \) |
| 29 | \( 1 + 140190 T + p^{7} T^{2} \) |
| 31 | \( 1 - 291208 T + p^{7} T^{2} \) |
| 37 | \( 1 + 135046 T + p^{7} T^{2} \) |
| 41 | \( 1 + 804438 T + p^{7} T^{2} \) |
| 43 | \( 1 + 721268 T + p^{7} T^{2} \) |
| 47 | \( 1 - 802656 T + p^{7} T^{2} \) |
| 53 | \( 1 - 274098 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1969440 T + p^{7} T^{2} \) |
| 61 | \( 1 - 3179342 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1363756 T + p^{7} T^{2} \) |
| 71 | \( 1 - 4389888 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4278862 T + p^{7} T^{2} \) |
| 79 | \( 1 + 3851960 T + p^{7} T^{2} \) |
| 83 | \( 1 + 8532228 T + p^{7} T^{2} \) |
| 89 | \( 1 - 3733410 T + p^{7} T^{2} \) |
| 97 | \( 1 + 15686206 T + p^{7} T^{2} \) |
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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
−10.27392822993227996570234374998, −9.626965835374019918314453054913, −8.505973370029222443213178220041, −7.10649114291352915601143545033, −6.42314088145121661104218751886, −5.39885695127981457476682257062, −3.86659993377490614303374824238, −3.09344711296225157961450274914, −1.12207210221798830130939198447, 0,
1.12207210221798830130939198447, 3.09344711296225157961450274914, 3.86659993377490614303374824238, 5.39885695127981457476682257062, 6.42314088145121661104218751886, 7.10649114291352915601143545033, 8.505973370029222443213178220041, 9.626965835374019918314453054913, 10.27392822993227996570234374998
