| L(s) = 1 | + 81·3-s + 1.81e3·5-s − 3.81e3·7-s + 6.56e3·9-s + 7.94e4·11-s + 1.54e5·13-s + 1.47e5·15-s + 3.43e5·17-s + 2.55e4·19-s − 3.09e5·21-s + 2.10e6·23-s + 1.35e6·25-s + 5.31e5·27-s + 1.13e6·29-s + 5.73e6·31-s + 6.43e6·33-s − 6.94e6·35-s − 9.67e6·37-s + 1.25e7·39-s − 1.08e7·41-s − 1.08e5·43-s + 1.19e7·45-s − 4.24e7·47-s − 2.57e7·49-s + 2.78e7·51-s − 7.58e7·53-s + 1.44e8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.30·5-s − 0.600·7-s + 0.333·9-s + 1.63·11-s + 1.49·13-s + 0.751·15-s + 0.997·17-s + 0.0448·19-s − 0.346·21-s + 1.56·23-s + 0.692·25-s + 0.192·27-s + 0.298·29-s + 1.11·31-s + 0.945·33-s − 0.781·35-s − 0.848·37-s + 0.865·39-s − 0.601·41-s − 0.00485·43-s + 0.433·45-s − 1.26·47-s − 0.638·49-s + 0.575·51-s − 1.32·53-s + 2.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.253512873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.253512873\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 81T \) |
| good | 5 | \( 1 - 1.81e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.81e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.94e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.54e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.43e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.55e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.10e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.13e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.73e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.67e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.08e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.08e5T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.24e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.58e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.39e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.38e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.57e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.75e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.83e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.68e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.19e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.32e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.37e9T + 7.60e17T^{2} \) |
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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
−9.579474967703543731333423594592, −9.129315160316635037825445899741, −8.206731208633674727842783870638, −6.61850662794234907328942965591, −6.37215697482368782477628354805, −5.13073777859526291827381865642, −3.71215699096809311808831218894, −3.01435676038635351805363577479, −1.58285604268101640001852821973, −1.08407537519477979429015482689,
1.08407537519477979429015482689, 1.58285604268101640001852821973, 3.01435676038635351805363577479, 3.71215699096809311808831218894, 5.13073777859526291827381865642, 6.37215697482368782477628354805, 6.61850662794234907328942965591, 8.206731208633674727842783870638, 9.129315160316635037825445899741, 9.579474967703543731333423594592
