| L(s) = 1 | + 57.6·5-s − 156.·7-s + 218.·11-s − 587.·13-s + 2.12e3·17-s − 29.7·19-s − 3.17e3·23-s + 196·25-s + 3.64e3·29-s − 6.33e3·31-s − 9.04e3·35-s − 8.43e3·37-s + 1.01e4·41-s + 6.85e3·43-s − 2.44e4·47-s + 7.80e3·49-s − 2.54e4·53-s + 1.25e4·55-s + 2.06e4·59-s + 4.55e4·61-s − 3.38e4·65-s − 1.90e4·67-s − 2.30e4·71-s − 8.96e4·73-s − 3.42e4·77-s + 3.40e4·79-s + 1.08e5·83-s + ⋯ |
| L(s) = 1 | + 1.03·5-s − 1.21·7-s + 0.543·11-s − 0.964·13-s + 1.78·17-s − 0.0189·19-s − 1.25·23-s + 0.0627·25-s + 0.804·29-s − 1.18·31-s − 1.24·35-s − 1.01·37-s + 0.942·41-s + 0.565·43-s − 1.61·47-s + 0.464·49-s − 1.24·53-s + 0.560·55-s + 0.774·59-s + 1.56·61-s − 0.994·65-s − 0.519·67-s − 0.542·71-s − 1.96·73-s − 0.657·77-s + 0.613·79-s + 1.72·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 57.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 156.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 218.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 587.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.12e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 29.7T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.85e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.54e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.10e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.88e4T + 8.58e9T^{2} \) |
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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
−9.741650737593794441124680791469, −9.427784943043412589671053629196, −8.028887794830515954187824487356, −6.94480675800215320951696465845, −6.06607211911680029085485565585, −5.32525073987130699553035504698, −3.80531618272082985293000594970, −2.75564064030339552825275996286, −1.51651300743688469540829050251, 0,
1.51651300743688469540829050251, 2.75564064030339552825275996286, 3.80531618272082985293000594970, 5.32525073987130699553035504698, 6.06607211911680029085485565585, 6.94480675800215320951696465845, 8.028887794830515954187824487356, 9.427784943043412589671053629196, 9.741650737593794441124680791469
